{"id":117750,"date":"2026-10-25T09:32:48","date_gmt":"2026-10-25T01:32:48","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/117750.html"},"modified":"2026-10-25T09:32:48","modified_gmt":"2026-10-25T01:32:48","slug":"2026-10-25%ef%bc%9a%e7%bb%9f%e8%ae%a1%e6%9c%89%e6%95%88%e5%ba%8f%e5%88%97%e6%95%b0%e7%9b%ae%e3%80%82%e7%94%a8go%e8%af%ad%e8%a8%80%ef%bc%8c%e7%bb%99%e5%ae%9a%e4%b8%a4%e4%b8%aa%e6%ad%a3%e6%95%b4","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/117750.html","title":{"rendered":"2026-10-25\uff1a\u7edf\u8ba1\u6709\u6548\u5e8f\u5217\u6570\u76ee\u3002\u7528go\u8bed\u8a00\uff0c\u7ed9\u5b9a\u4e24\u4e2a\u6b63\u6574\u6570 n \u548c k\u3002\u73b0\u5728\u8981\u7edf\u8ba1\u6ee1\u8db3\u4e0b\u9762\u6761\u4ef6\u7684\u6709\u5e8f\u5e8f\u5217\u6570\u91cf\uff1a\u5e8f\u5217\u4e00\u5171\u5305\u542b k \u4e2a\u6b63\u6574\u6570\uff0c\u8fd9\u4e9b\u6570\u7684\u603b\u548c\u6070\u597d\u4e3a n\uff0c\u5e76\u4e14\u8fd9\u4e9b\u6570\u7684\u4e58\u79ef\u5fc5\u987b\u662f\u5076\u6570\u3002\u5e8f"},"content":{"rendered":"

2026-10-25：\u7edf\u8ba1\u6709\u6548\u5e8f\u5217\u6570\u76ee\u3002\u7528go\u8bed\u8a00，\u7ed9\u5b9a\u4e24\u4e2a\u6b63\u6574\u6570 n \u548c k\u3002\u73b0\u5728\u8981\u7edf\u8ba1\u6ee1\u8db3\u4e0b\u9762\u6761\u4ef6\u7684\u6709\u5e8f\u5e8f\u5217\u6570\u91cf：\u5e8f\u5217\u4e00\u5171\u5305\u542b k \u4e2a\u6b63\u6574\u6570，\u8fd9\u4e9b\u6570\u7684\u603b\u548c\u6070\u597d\u4e3a n，\u5e76\u4e14\u8fd9\u4e9b\u6570\u7684\u4e58\u79ef\u5fc5\u987b\u662f\u5076\u6570\u3002\u5e8f\u5217\u6309\u4f4d\u7f6e\u533a\u5206，\u53ea\u8981\u67d0\u4e2a\u4f4d\u7f6e\u4e0a\u7684\u6570\u4e0d\u540c，\u5c31\u7b97\u4f5c\u4e0d\u540c\u5e8f\u5217\u3002\u6700\u7ec8\u7ed3\u679c\u9700\u8981\u5bf9 1000000007 \u53d6\u4f59\u540e\u8fd4\u56de\u3002<\/p>\n

1 <= n <= 500000\u3002<\/p>\n

1 <= k <= n\u3002<\/p>\n

\u8f93\u5165： n = 5, k = 3\u3002<\/p>\n

\u8f93\u51fa： 3\u3002<\/p>\n

\u89e3\u91ca：<\/p>\n

\u957f\u5ea6\u4e3a k = 3 \u4e14\u548c\u4e3a 5 \u7684\u5e8f\u5217\u6709：<\/p>\n\n\u5e8f\u5217\u4e58\u79ef\u5947\u5076\u6027<\/tr>\n\n\n\n\n\n\n\n
[1, 1, 3]<\/td>\n1 * 1 * 3 = 3<\/td>\n\u5947\u6570<\/td>\n<\/tr>\n
[1, 2, 2]<\/td>\n1 * 2 * 2 = 4<\/td>\n\u5076\u6570<\/td>\n<\/tr>\n
[2, 1, 2]<\/td>\n2 * 1 * 2 = 4<\/td>\n\u5076\u6570<\/td>\n<\/tr>\n
[2, 2, 1]<\/td>\n2 * 2 * 1 = 4<\/td>\n\u5076\u6570<\/td>\n<\/tr>\n
[1, 3, 1]<\/td>\n1 * 3 * 1 = 3<\/td>\n\u5947\u6570<\/td>\n<\/tr>\n
[3, 1, 1]<\/td>\n3 * 1 * 1 = 3<\/td>\n\u5947\u6570<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u6709 3 \u4e2a\u5e8f\u5217\u7684\u4e58\u79ef\u662f\u5076\u6570，\u56e0\u6b64\u7b54\u6848\u662f 3\u3002<\/p>\n

\u9898\u76ee\u6765\u81ea\u529b\u62634002\u3002<\/p>\n

\u5927\u4f53\u6b65\u9aa4\u5982\u4e0b：<\/h2>\n

\u6b65\u9aa4 1：\u628a\u9898\u76ee\u7ffb\u8bd1\u6210\u7ec4\u5408\u8ba1\u6570<\/h3>\n
    \n
  • \u5e8f\u5217\u6709\u5e8f\u3001\u6709 k \u4e2a\u4f4d\u7f6e\u3001\u6bcf\u4e2a\u4f4d\u7f6e\u662f \u22651 \u7684\u6574\u6570\u3001\u603b\u548c\u6070\u597d\u4e3a n \u2014\u2014 \u8fd9\u5728\u7ec4\u5408\u6570\u5b66\u91cc\u5c31\u662f n \u7684 k \u90e8\u5206\u6709\u5e8f\u5206\u62c6（composition）\u3002<\/li>\n
  • \u4f4d\u7f6e\u533a\u5206\u610f\u5473\u7740 [1,2,2]\u3001[2,1,2]\u3001[2,2,1] \u662f\u4e09\u4e2a\u4e0d\u540c\u5e8f\u5217，\u53ef\u4ee5\u76f4\u63a5\u7528\u300c\u628a n \u4e2a 1 \u6392\u6210\u4e00\u6392\u3001\u5728 n\u22121 \u4e2a\u7f1d\u9699\u91cc\u63d2 k\u22121 \u4e2a\u9694\u677f\u300d\u6765\u6570，\u5929\u7136\u5e26\u987a\u5e8f，\u4e0d\u9700\u8981\u989d\u5916\u4e58\u6392\u5217\u6570\u3002<\/li>\n<\/ul>\n

    \u6b65\u9aa4 2：\u6b63\u96be\u5219\u53cd \u2014\u2014 \u7528\u8865\u96c6<\/h3>\n

    \u300c\u4e58\u79ef\u662f\u5076\u6570\u300d\u7b49\u4ef7\u4e8e\u300c\u81f3\u5c11\u6709\u4e00\u4e2a\u6570\u662f\u5076\u6570\u300d\u3002\u76f4\u63a5\u6570\u300c\u81f3\u5c11\u4e00\u4e2a\u300d\u8981\u5206 1 \u4e2a\u5076\u6570\u30012 \u4e2a\u5076\u6570\u2026\u2026\u4f1a\u9677\u5165\u5bb9\u65a5，\u5f88\u9ebb\u70e6\u3002<\/p>\n

    \u53cd\u8fc7\u6765\u60f3：<\/p>\n

    \u4e58\u79ef\u662f\u5076\u6570 = \u603b\u6570 \u2212 \u4e58\u79ef\u662f\u5947\u6570<\/p>\n

    \u800c\u300c\u4e58\u79ef\u662f\u5947\u6570\u300d\u7684\u5145\u8981\u6761\u4ef6\u662f\u300ck \u4e2a\u6570\u5168\u90e8\u662f\u5947\u6570\u300d\u2014\u2014\u8fd9\u4e2a\u6761\u4ef6\u975e\u5e38\u5f3a\u3001\u975e\u5e38\u597d\u6570\u3002\u4e8e\u662f\u95ee\u9898\u88ab\u7b80\u5316\u4e3a\u4e24\u4e2a\u7eaf\u7ec4\u5408\u6570\u76f8\u51cf\u3002<\/p>\n

    \u6b65\u9aa4 3：\u6570\u603b\u6570（\u9694\u677f\u6cd5 \/ \u661f\u4e0e\u68d2）<\/h3>\n

    \u628a n \u5199\u6210 n \u4e2a 1：<\/p>\n

    1 1 1 … 1 （n \u4e2a）<\/p>\n

    \u4e2d\u95f4\u6709 n\u22121 \u4e2a\u7a7a\u9699，\u9009 k\u22121 \u4e2a\u7a7a\u9699\u63d2\u677f，\u5c31\u628a n \u5207\u6210 k \u6bb5，\u6bcf\u6bb5\u81f3\u5c11 1 \u4e2a 1，\u6bb5\u957f\u5c31\u662f\u5e8f\u5217\u7684\u6bcf\u4e00\u9879\u3002<\/p>\n

    \u603b\u6570=(n\u22121k\u22121)\\\\text{\u603b\u6570} = \\\\binom{n-1}{k-1}<\/span><\/span>\u603b\u6570<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span><\/span><\/span>k<\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span><\/span><\/span>n<\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u4ee3\u7801\u91cc\u5c31\u662f comb(n-1, k-1)\u3002<\/p>\n

    \u6b65\u9aa4 4：\u6570\u300c\u5168\u662f\u5947\u6570\u300d\u7684\u5e8f\u5217（\u6362\u5143 + \u53ef\u884c\u6027\u5224\u65ad）<\/h3>\n

    \u8bbe\u7b2c i \u9879\u4e3a ai=2bi+1a_i = 2b_i + 1<\/span><\/span>a<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span>b<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>，\u5176\u4e2d bi\u22650b_i \\\\ge 0<\/span><\/span>b<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2265<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>（\u8fd9\u6837 aia_i<\/span><\/span>a<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u81ea\u52a8\u662f\u6b63\u5947\u6570）\u3002<\/p>\n

    \u6c42\u548c：<\/p>\n

    \u2211ai=2\u2211bi+k=n\u27f9\u2211i=1kbi=n\u2212k2\\\\sum a_i = 2\\\\sum b_i + k = n \\\\quad\\\\Longrightarrow\\\\quad \\\\sum_{i=1}^{k} b_i = \\\\frac{n-k}{2}<\/span><\/span>\u2211<\/span><\/span>a<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span>\u2211<\/span><\/span>b<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>k<\/span><\/span>=<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span>\u27f9<\/span><\/span><\/span><\/span><\/span><\/span>i<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span>\u2212<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u8fd9\u91cc\u51fa\u73b0\u7b2c\u4e00\u4e2a\u5173\u952e\u5224\u5b9a：<\/p>\n

      \n
    • n\u2212k2\\\\frac{n-k}{2}<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>\u2212<\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5fc5\u987b\u662f\u975e\u8d1f\u6574\u6570\u3002<\/li>\n
    • \u975e\u8d1f\u6027：\u9898\u76ee\u7ed9\u4e86 k\u2264nk \\\\le n<\/span><\/span>k<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>，\u6240\u4ee5 n\u2212k\u22650n-k\\\\ge0<\/span><\/span>n<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>k<\/span><\/span>\u2265<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span> \u6052\u6210\u7acb，\u4e0d\u7528\u7ba1\u3002<\/li>\n
    • \u6574\u6570\u6027：\u9700\u8981 n\u2212kn-k<\/span><\/span>n<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u4e3a\u5076\u6570\u3002\u7531\u4e8e n\u2212kn-k<\/span><\/span>n<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u4e0e n+kn+k<\/span><\/span>n<\/span><\/span>+<\/span><\/span><\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u5947\u5076\u6027\u76f8\u540c，\u4ee3\u7801\u91cc\u5199\u6210 (n+k)%2 == 0\u3002<\/li>\n
    • \u5982\u679c\u4e0d\u6ee1\u8db3，\u8bf4\u660e\u6839\u672c\u51d1\u4e0d\u51fa\u300c\u5168\u5947\u6570\u300d\u5e8f\u5217，\u51cf\u6cd5\u9879\u4e3a 0，\u7b54\u6848\u5c31\u662f\u603b\u6570\u3002<\/li>\n<\/ul>\n

      \u6ee1\u8db3\u65f6，\u95ee\u9898\u53d8\u6210\u300ck \u4e2a\u975e\u8d1f\u6574\u6570 bib_i<\/span><\/span>b<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e4b\u548c\u4e3a m=n\u2212k2m=\\\\frac{n-k}{2}<\/span><\/span>m<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>\u2212<\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u300d\u7684\u7ecf\u5178\u7403\u76d2\u95ee\u9898，\u9694\u677f\u6cd5（\u5141\u8bb8\u7a7a\u76d2）\u7ed9\u51fa：<\/p>\n

      (m+k\u22121k\u22121)=(n\u2212k2+k\u22121k\u22121)=(n+k2\u22121k\u22121)\\\\binom{m+k-1}{k-1} = \\\\binom{\\\\frac{n-k}{2}+k-1}{k-1} = \\\\binom{\\\\frac{n+k}{2}-1}{k-1}<\/span><\/span>(<\/span><\/span><\/span>k<\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span><\/span><\/span>m<\/span><\/span>+<\/span><\/span>k<\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span><\/span><\/span>k<\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>\u2212<\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>k<\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span><\/span><\/span>k<\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>+<\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

      \u4ee3\u7801\u91cc\u6b63\u662f comb((n+k)\/2-1, k-1)\u3002\u6ce8\u610f Go \u91cc\u6574\u6570\u9664\u6cd5\u5929\u7136\u5411\u4e0b\u53d6\u6574，\u800c\u8fd9\u4e2a\u5206\u652f\u53ea\u5728 n+k \u4e3a\u5076\u6570\u65f6\u624d\u8fdb\u5165，\u6240\u4ee5\u4e0d\u4f1a\u53d6\u9519\u3002<\/p>\n

      \u6b65\u9aa4 5：\u5408\u8d77\u6765\u5f97\u5230\u95ed\u5f0f\u89e3<\/h3>\n

      ans=(n\u22121k\u22121)\u2005\u200a\u2212\u2005\u200a[(n+k)\u00a0\u4e3a\u5076\u6570]\u22c5(n+k2\u22121k\u22121)\\\\text{ans} = \\\\binom{n-1}{k-1} \\\\;-\\\\; \\\\big[(n+k)\\\\text{ \u4e3a\u5076\u6570}\\\\big]\\\\cdot \\\\binom{\\\\frac{n+k}{2}-1}{k-1}<\/span><\/span>ans<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span><\/span><\/span>k<\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span><\/span><\/span>n<\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span>[<\/span><\/span>(<\/span>n<\/span><\/span>+<\/span><\/span><\/span><\/span>k<\/span>)<\/span>\u00a0<\/span>\u4e3a\u5076\u6570<\/span><\/span>]<\/span><\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>(<\/span><\/span><\/span>k<\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>+<\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

      \u4ee3\u7801\u91cc\u90a3\u4e00\u53e5 + mod \u662f\u9632\u8d1f\u6570：\u4e24\u4e2a\u7ec4\u5408\u6570\u5404\u81ea\u90fd\u5728 [0,mod)[0, \\\\text{mod})<\/span><\/span>[<\/span>0<\/span>,<\/span><\/span>mod<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span> \u5185，\u76f8\u51cf\u53ef\u80fd\u843d\u5230\u8d1f\u6570\u533a\u95f4，\u800c Go \u7684 % \u5bf9\u8d1f\u6570\u4f1a\u8fd4\u56de\u8d1f\u4f59\u6570，\u6240\u4ee5\u5148\u52a0\u4e00\u4e2a mod \u518d\u53d6\u6a21，\u4fdd\u8bc1\u7ed3\u679c\u843d\u5728 [0,mod)[0, \\\\text{mod})<\/span><\/span>[<\/span>0<\/span>,<\/span><\/span>mod<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

      （\u53ef\u4ee5\u9a8c\u8bc1\u51cf\u6cd5\u9879\u6c38\u8fdc\u4e0d\u8d85\u8fc7\u603b\u6570：\u5168\u5947\u6570\u5e8f\u5217\u662f\u603b\u6570\u7684\u4e00\u4e2a\u5b50\u96c6，\u6240\u4ee5\u5dee\u503c\u5fc5\u7136 \u2265 0，\u52a0\u4e00\u6b21 mod \u8db3\u591f\u3002）<\/p>\n

      \u6b65\u9aa4 6：\u6a21\u610f\u4e49\u4e0b\u5982\u4f55 O(1) \u7b97\u7ec4\u5408\u6570<\/h3>\n

      \u6a21\u6570 109+710^9+7<\/span><\/span>1<\/span>0<\/span><\/span>9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>7<\/span><\/span><\/span><\/span><\/span> \u662f\u8d28\u6570，\u4e14\u8fdc\u5927\u4e8e\u9898\u76ee\u4e2d\u53ef\u80fd\u51fa\u73b0\u7684\u6700\u5927\u4e0b\u6807（n\u22121 \u2264 499999），\u56e0\u6b64：<\/p>\n

        \n
      • \u6240\u6709 1,2,\u2026,4999991,2,\\\\dots,499999<\/span><\/span>1<\/span>,<\/span><\/span>2<\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>499999<\/span><\/span><\/span><\/span><\/span> \u90fd\u4e0e mod \u4e92\u8d28，\u9636\u4e58\u53d6\u6a21\u540e\u975e\u96f6\u3001\u53ef\u9006；<\/li>\n
      • \u53ef\u4ee5\u7528\u8d39\u9a6c\u5c0f\u5b9a\u7406\u6c42\u9006：x\u22121\u2261xmod\u22122(modmod)x^{-1} \\\\equiv x^{\\\\text{mod}-2} \\\\pmod{\\\\text{mod}}<\/span><\/span>x<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2261<\/span><\/span><\/span><\/span>x<\/span><\/span>mod<\/span><\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>mod<\/span><\/span><\/span><\/span>mod<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/li>\n<\/ul>\n

        \u4ee3\u7801\u5728 init() \u91cc\u505a\u4e86\u4e24\u4ef6\u4e8b：<\/p>\n

      • \u6b63\u5411\u9012\u63a8\u9636\u4e58：fac[i] = fac[i-1] * i % mod，\u5f97\u5230 fac[i] = i!\u3002<\/li>\n
      • \u53cd\u5411\u9012\u63a8\u9636\u4e58\u9006\u5143：\u5148\u82b1\u4e00\u6b21\u5feb\u901f\u5e42\u7b97\u51fa invF[mx-1] = (499999!)^{-1}，\u518d\u4ece\u540e\u5f80\u524d\u7528
        \n1(i\u22121)!=ii!=invF[i]\u22c5i\\\\frac{1}{(i-1)!} = \\\\frac{i}{i!} = \\\\text{invF}[i]\\\\cdot i<\/span><\/span><\/span><\/span>(<\/span>i<\/span><\/span>\u2212<\/span><\/span>1<\/span>)!<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>i<\/span>!<\/span><\/span><\/span><\/span><\/span><\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>invF<\/span><\/span>[<\/span>i<\/span>]<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>i<\/span><\/span><\/span><\/span><\/span><\/span>
        \n\u9012\u63a8\u51fa\u6240\u6709 invF[i]\u3002<\/li>\n

        \u8fd9\u4e2a\u300c\u4e00\u6b21\u5feb\u901f\u5e42 + \u7ebf\u6027\u56de\u586b\u300d\u7684\u6280\u5de7，\u628a k \u6b21（\u751a\u81f3 5\u00d710\u2075 \u6b21）\u5feb\u901f\u5e42\u538b\u6210\u4e86 1 \u6b21，\u662f\u8fd9\u91cc\u7684\u6838\u5fc3\u4f18\u5316\u70b9\u3002<\/p>\n

        \u6b65\u9aa4 7：comb \u5c31\u662f\u67e5\u8868<\/h3>\n

        (nm)=n!m!\u2009(n\u2212m)!=fac[n]\u22c5invF[m]\u22c5invF[n\u2212m]\u200amod\u200amod\\\\binom{n}{m} = \\\\frac{n!}{m!\\\\,(n-m)!} = \\\\text{fac}[n]\\\\cdot \\\\text{invF}[m]\\\\cdot \\\\text{invF}[n-m] \\\\bmod \\\\text{mod}<\/span><\/span>(<\/span><\/span><\/span>m<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>!<\/span><\/span>(<\/span>n<\/span><\/span>\u2212<\/span><\/span>m<\/span>)!<\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>!<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>fac<\/span><\/span>[<\/span>n<\/span>]<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>invF<\/span><\/span>[<\/span>m<\/span>]<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>invF<\/span><\/span>[<\/span>n<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>m<\/span>]<\/span><\/span><\/span>mod<\/span><\/span><\/span><\/span><\/span><\/span><\/span>mod<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

        \u4e09\u6b21\u6570\u7ec4\u8bbf\u95ee + \u4e24\u6b21\u4e58\u6cd5\u53d6\u6a21，O(1)\u3002\u9898\u76ee\u7ea6\u675f\u4e0b\u6240\u6709\u4e0b\u6807\u90fd\u5408\u6cd5：<\/p>\n

          \n
        • n\u22121\u2264499999=mx\u22121n-1 \\\\le 499999 = \\\\text{mx}-1<\/span><\/span>n<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>499999<\/span><\/span>=<\/span><\/span><\/span><\/span>mx<\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>；<\/li>\n
        • n+k2\u22121\u2264n\u22121\\\\frac{n+k}{2}-1 \\\\le n-1<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>+<\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>n<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>（\u56e0\u4e3a k\u2264nk\\\\le n<\/span><\/span>k<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>），\u4e5f\u5728\u8868\u5185；<\/li>\n
        • \u4e14 n+k2\u22121\u2265k\u22121\\\\frac{n+k}{2}-1 \\\\ge k-1<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>+<\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span><\/span>\u2265<\/span><\/span><\/span><\/span>k<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>（\u56e0\u4e3a n\u2265kn\\\\ge k<\/span><\/span>n<\/span><\/span>\u2265<\/span><\/span><\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>），\u4e0d\u4f1a\u51fa\u73b0\u4e0a\u6807\u5c0f\u4e8e\u4e0b\u6807\u3002<\/li>\n<\/ul>\n

          \u6b65\u9aa4 8：\u4ee3\u5165\u6837\u4f8b n=5, k=3 \u8d70\u4e00\u904d<\/h3>\n
            \n
          • \u603b\u6570：(42)=6\\\\binom{4}{2} = 6<\/span><\/span>(<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span>，\u5bf9\u5e94\u4f60\u8868\u683c\u91cc\u90a3 6 \u884c\u3002<\/li>\n
          • n+k = 8 \u4e3a\u5076\u6570，\u9700\u8981\u51cf\u5168\u5947\u6570\u7684\u60c5\u51b5：n\u2212k2=1\\\\frac{n-k}{2} = 1<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>\u2212<\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>，\u5373 b1+b2+b3=1b_1+b_2+b_3=1<\/span><\/span>b<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>，\u89e3\u6570 (1+3\u221213\u22121)=(32)=3\\\\binom{1+3-1}{3-1}=\\\\binom{3}{2}=3<\/span><\/span>(<\/span><\/span><\/span>3<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>1<\/span>+<\/span>3<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>，\u6b63\u597d\u5bf9\u5e94 [1,1,3]\u3001[1,3,1]\u3001[3,1,1] \u8fd9 3 \u884c\u3002<\/li>\n
          • \u7b54\u6848：6 \u2212 3 = 3，\u4e0e\u9898\u76ee\u8f93\u51fa\u4e00\u81f4\u3002<\/li>\n<\/ul>\n

            \u6b65\u9aa4 9：\u4e24\u4e2a\u8fb9\u754c\u81ea\u68c0（\u5e2e\u52a9\u786e\u8ba4\u516c\u5f0f\u6ca1\u9519）<\/h3>\n
              \n
            • k = 1：\u5e8f\u5217\u53ea\u6709 [n]，\u4e58\u79ef\u4e3a\u5076\u6570\u5f53\u4e14\u4ec5\u5f53 n \u4e3a\u5076\u6570\u3002\u516c\u5f0f\u7ed9\u51fa (00)=1\\\\binom{0}{0}=1<\/span><\/span>(<\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>，n \u4e3a\u5947\u6570\u65f6\u51cf (\u22c50)=1\\\\binom{\\\\cdot}{0}=1<\/span><\/span>(<\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u22c5<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> \u5f97 0，n \u4e3a\u5076\u6570\u65f6\u4e0d\u51cf\u5f97 1\u3002\u2714<\/li>\n
            • k = n：\u6bcf\u9879\u53ea\u80fd\u662f 1，\u4e58\u79ef\u6052\u4e3a 1（\u5947\u6570）\u3002\u516c\u5f0f\u7ed9\u51fa 1\u2212(n\u22121n\u22121)=01 – \\\\binom{n-1}{n-1} = 0<\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>(<\/span><\/span><\/span>n<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>n<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u3002\u2714<\/li>\n<\/ul>\n
              \n

              \u590d\u6742\u5ea6<\/h3>\n\n\u9879\u76ee\u7ed3\u8bba\u8bf4\u660e<\/tr>\n\n\n\n\n\n
              \u9884\u5904\u7406\u65f6\u95f4<\/td>\nO(mx+log\u2061mod)\\\\mathcal{O}(\\\\text{mx} + \\\\log \\\\text{mod})<\/span><\/span>O<\/span>(<\/span>mx<\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>log<\/span><\/span><\/span>mod<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/td>\n\u9636\u4e58\u6b63\u5411\u9012\u63a8 5\u00d710\u2075 \u6b21；\u9636\u4e58\u9006\u5143\u53cd\u5411\u9012\u63a8 5\u00d710\u2075 \u6b21；\u5916\u52a0\u4e00\u6b21\u5feb\u901f\u5e42，\u6307\u6570\u4e3a\u5e38\u6570 109+510^9+5<\/span><\/span>1<\/span>0<\/span><\/span>9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/span>，\u7ea6 30 \u6b21\u4e58\u65b9\u53d6\u6a21（\u5373 O(log\u2061mod)\\\\mathcal{O}(\\\\log \\\\text{mod})<\/span><\/span>O<\/span>(<\/span>log<\/span><\/span><\/span>mod<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>，\u53ef\u89c6\u4e3a\u5e38\u6570）<\/td>\n<\/tr>\n
              \u5355\u6b21\u67e5\u8be2\u65f6\u95f4<\/td>\nO(1)\\\\mathcal{O}(1)<\/span><\/span>O<\/span>(<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/td>\n\u5224\u65ad\u5947\u5076 + \u4e24\u6b21 comb（\u5404 3 \u6b21\u6570\u7ec4\u8bbf\u95ee\u30012 \u6b21\u4e58\u6cd5\u53d6\u6a21）<\/td>\n<\/tr>\n
              \u603b\u65f6\u95f4\u590d\u6742\u5ea6<\/td>\nO(mx)=O(5\u00d7105)\\\\mathcal{O}(\\\\text{mx}) = \\\\mathcal{O}(5\\\\times10^{5})<\/span><\/span>O<\/span>(<\/span>mx<\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>O<\/span>(<\/span>5<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>1<\/span>0<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>，\u66f4\u4e25\u8c28\u5199\u4e3a O(mx+log\u2061mod)\\\\mathcal{O}(\\\\text{mx}+\\\\log\\\\text{mod})<\/span><\/span>O<\/span>(<\/span>mx<\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>log<\/span><\/span><\/span>mod<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/td>\n\u4e0e n\u3001k \u65e0\u5173，\u662f\u5e38\u6570\u7ea7\u9884\u5904\u7406 + \u5e38\u6570\u7ea7\u67e5\u8be2<\/td>\n<\/tr>\n
              \u603b\u989d\u5916\u7a7a\u95f4\u590d\u6742\u5ea6<\/td>\nO(mx)\\\\mathcal{O}(\\\\text{mx})<\/span><\/span>O<\/span>(<\/span>mx<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/td>\n\u4e24\u4e2a\u957f\u5ea6 5\u00d710\u2075 \u7684\u5168\u5c40\u6570\u7ec4 fac \u4e0e invF\u3002Go \u5728 64 \u4f4d\u5e73\u53f0\u4e0a int \u4e3a 8 \u5b57\u8282，\u5408\u8ba1 2\u00d75\u00d7105\u00d78\u00a0B\u22488\u00a0MB2\\\\times 5\\\\times10^{5}\\\\times 8\\\\ \\\\text{B} \\\\approx 8\\\\ \\\\text{MB}<\/span><\/span>2<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>5<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>1<\/span>0<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>8<\/span>\u00a0<\/span>B<\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>8<\/span>\u00a0<\/span>MB<\/span><\/span><\/span><\/span><\/span><\/span>（\u7ea6 7.63 MiB），\u5176\u4f59\u53ea\u6709\u51e0\u4e2a\u6807\u91cf\u53d8\u91cf，\u4e3a O(1)\\\\mathcal{O}(1)<\/span><\/span>O<\/span>(<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

              \u503c\u5f97\u4e00\u63d0\u7684\u51e0\u4e2a\u70b9<\/h3>\n
                \n
              • \u7a7a\u95f4\u6362\u65f6\u95f4：\u5982\u679c\u6bcf\u6b21\u67e5\u8be2\u90fd\u73b0\u7b97\u9636\u4e58，\u662f O(n)\\\\mathcal{O}(n)<\/span><\/span>O<\/span>(<\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span> \u4e00\u6b21；\u9884\u5904\u7406\u540e\u53d8\u6210 O(1)\\\\mathcal{O}(1)<\/span><\/span>O<\/span>(<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span> \u4e00\u6b21，\u4ee3\u4ef7\u662f\u7ea6 8 MB \u5e38\u9a7b\u5185\u5b58\u3002\u5bf9 n \u4e0a\u9650 5\u00d710\u2075 \u6765\u8bf4\u8fd9\u4e2a\u4ea4\u6362\u975e\u5e38\u5212\u7b97\u3002<\/li>\n
              • mx \u7684\u53d6\u503c\u521a\u597d\u5361\u6ee1：mx = 500000 \u65f6\u6700\u5927\u53ef\u7528\u4e0b\u6807\u662f 499999，\u6b63\u597d\u7b49\u4e8e n\u22121 \u7684\u4e0a\u9650；\u5982\u679c n \u518d\u5927\u4e00\u70b9\u5c31\u4f1a\u8d8a\u754c，\u6240\u4ee5\u6570\u7ec4\u5927\u5c0f\u548c\u9898\u76ee\u7ea6\u675f\u662f\u7ed1\u5b9a\u7684\u3002<\/li>\n
              • \u8d39\u9a6c\u9006\u5143\u6210\u7acb\u7684\u524d\u63d0\u662f mod \u4e3a\u8d28\u6570\u4e14\u5927\u4e8e\u6240\u6709\u53c2\u4e0e\u8fd0\u7b97\u7684\u6574\u6570，1e9+7 \u4e0e 5\u00d710\u2075 \u7684\u7ec4\u5408\u5b8c\u5168\u6ee1\u8db3；\u82e5\u6362\u6210\u975e\u8d28\u6570\u6a21\u6216\u4e0e n \u540c\u91cf\u7ea7\u7684\u5c0f\u6a21\u6570，\u8fd9\u5957\u9636\u4e58\u9006\u5143\u5c31\u4f1a\u5931\u6548（fac \u53ef\u80fd\u53d8\u6210 0 \u800c\u4e0d\u53ef\u9006）\u3002<\/li>\n<\/ul>\n

                Go\u5b8c\u6574\u4ee3\u7801\u5982\u4e0b：<\/h2>\n

                package<\/span> main<\/p>\n

                import<\/span> (<\/span>
                \n"fmt"<\/span>
                \n)<\/span><\/p>\n

                const<\/span> mod =<\/span> 1_000_000_007<\/span>
                \nconst<\/span> mx =<\/span> 500_000<\/span><\/p>\n

                var<\/span> fac [<\/span>mx]<\/span>int<\/span> \/\/ fac[i] = i!<\/span>
                \nvar<\/span> invF [<\/span>mx]<\/span>int<\/span> \/\/ invF[i] = i!^-1 = pow(i!, mod-2)<\/span><\/p>\n

                func<\/span> init<\/span>(<\/span>)<\/span> {<\/span>
                \nfac[<\/span>0<\/span>]<\/span> =<\/span> 1<\/span>
                \nfor<\/span> i :=<\/span> 1<\/span>;<\/span> i <<\/span> mx;<\/span> i++<\/span> {<\/span>
                \nfac[<\/span>i]<\/span> =<\/span> fac[<\/span>i–<\/span>1<\/span>]<\/span> *<\/span> i %<\/span> mod
                \n}<\/span><\/p>\n

                invF[<\/span>mx–<\/span>1<\/span>]<\/span> =<\/span> pow<\/span>(<\/span>fac[<\/span>mx–<\/span>1<\/span>]<\/span>,<\/span> mod–<\/span>2<\/span>)<\/span>
                \nfor<\/span> i :=<\/span> mx –<\/span> 1<\/span>;<\/span> i ><\/span> 0<\/span>;<\/span> i—<\/span> {<\/span>
                \ninvF[<\/span>i–<\/span>1<\/span>]<\/span> =<\/span> invF[<\/span>i]<\/span> *<\/span> i %<\/span> mod
                \n}<\/span>
                \n}<\/span><\/p>\n

                func<\/span> pow<\/span>(<\/span>x,<\/span> n int<\/span>)<\/span> int<\/span> {<\/span>
                \nres :=<\/span> 1<\/span>
                \nfor<\/span> ;<\/span> n ><\/span> 0<\/span>;<\/span> n \/=<\/span> 2<\/span> {<\/span>
                \nif<\/span> n%<\/span>2<\/span> ><\/span> 0<\/span> {<\/span>
                \nres =<\/span> res *<\/span> x %<\/span> mod
                \n}<\/span>
                \nx =<\/span> x *<\/span> x %<\/span> mod
                \n}<\/span>
                \nreturn<\/span> res
                \n}<\/span><\/p>\n

                \/\/ \u4ece n \u4e2a\u6570\u4e2d\u9009 m \u4e2a\u6570\u7684\u65b9\u6848\u6570<\/span>
                \nfunc<\/span> comb<\/span>(<\/span>n,<\/span> m int<\/span>)<\/span> int<\/span> {<\/span>
                \nreturn<\/span> fac[<\/span>n]<\/span> *<\/span> invF[<\/span>m]<\/span> %<\/span> mod *<\/span> invF[<\/span>n–<\/span>m]<\/span> %<\/span> mod
                \n}<\/span><\/p>\n

                func<\/span> countValidSequences<\/span>(<\/span>n,<\/span> k int<\/span>)<\/span> int<\/span> {<\/span>
                \nans :=<\/span> comb<\/span>(<\/span>n–<\/span>1<\/span>,<\/span> k–<\/span>1<\/span>)<\/span>
                \nif<\/span> (<\/span>n+<\/span>k)<\/span>%<\/span>2<\/span> ==<\/span> 0<\/span> {<\/span>
                \nans =<\/span> (<\/span>ans –<\/span> comb<\/span>(<\/span>(<\/span>n+<\/span>k)<\/span>\/<\/span>2<\/span>–<\/span>1<\/span>,<\/span> k–<\/span>1<\/span>)<\/span> +<\/span> mod)<\/span> %<\/span> mod \/\/ +mod \u4fdd\u8bc1\u7b54\u6848\u975e\u8d1f<\/span>
                \n}<\/span>
                \nreturn<\/span> ans
                \n}<\/span><\/p>\n

                func<\/span> main<\/span>(<\/span>)<\/span> {<\/span>
                \nn :=<\/span> 5<\/span>
                \nk :=<\/span> 3<\/span>
                \nresult :=<\/span> countValidSequences<\/span>(<\/span>n,<\/span> k)<\/span>
                \nfmt.<\/span>Println<\/span>(<\/span>result)<\/span>
                \n}<\/span><\/p>\n

                \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"<\/p>\n

                Python\u5b8c\u6574\u4ee3\u7801\u5982\u4e0b：<\/h2>\n

                # -*-coding:utf-8-*-<\/span><\/p>\n

                MOD =<\/span> 1_000_000_007<\/span>
                \nMX =<\/span> 500_000<\/span><\/p>\n

                fac =<\/span> [<\/span>1<\/span>]<\/span> *<\/span> MX # fac[i] = i!<\/span>
                \ninvF =<\/span> [<\/span>1<\/span>]<\/span> *<\/span> MX # invF[i] = i!^(-1) = pow(i!, MOD-2)<\/span><\/p>\n

                def<\/span> init<\/span>(<\/span>)<\/span>:<\/span>
                \n fac[<\/span>0<\/span>]<\/span> =<\/span> 1<\/span>
                \n for<\/span> i in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> MX)<\/span>:<\/span>
                \n fac[<\/span>i]<\/span> =<\/span> fac[<\/span>i –<\/span> 1<\/span>]<\/span> *<\/span> i %<\/span> MOD<\/p>\n

                invF[<\/span>MX –<\/span> 1<\/span>]<\/span> =<\/span> pow<\/span>(<\/span>fac[<\/span>MX –<\/span> 1<\/span>]<\/span>,<\/span> MOD –<\/span> 2<\/span>,<\/span> MOD)<\/span>
                \n for<\/span> i in<\/span> range<\/span>(<\/span>MX –<\/span> 1<\/span>,<\/span> 0<\/span>,<\/span> –<\/span>1<\/span>)<\/span>:<\/span>
                \n invF[<\/span>i –<\/span> 1<\/span>]<\/span> =<\/span> invF[<\/span>i]<\/span> *<\/span> i %<\/span> MOD<\/p>\n

                init(<\/span>)<\/span><\/p>\n

                # \u4ece n \u4e2a\u6570\u4e2d\u9009 m \u4e2a\u6570\u7684\u65b9\u6848\u6570<\/span>
                \ndef<\/span> comb<\/span>(<\/span>n,<\/span> m)<\/span>:<\/span>
                \n if<\/span> m <<\/span> 0<\/span> or<\/span> m ><\/span> n:<\/span>
                \n return<\/span> 0<\/span>
                \n return<\/span> fac[<\/span>n]<\/span> *<\/span> invF[<\/span>m]<\/span> %<\/span> MOD *<\/span> invF[<\/span>n –<\/span> m]<\/span> %<\/span> MOD<\/p>\n

                def<\/span> count_valid_sequences<\/span>(<\/span>n,<\/span> k)<\/span>:<\/span>
                \n ans =<\/span> comb(<\/span>n –<\/span> 1<\/span>,<\/span> k –<\/span> 1<\/span>)<\/span><\/p>\n

                if<\/span> (<\/span>n +<\/span> k)<\/span> %<\/span> 2<\/span> ==<\/span> 0<\/span>:<\/span>
                \n ans =<\/span> (<\/span>ans –<\/span> comb(<\/span>(<\/span>n +<\/span> k)<\/span> \/\/<\/span> 2<\/span> –<\/span> 1<\/span>,<\/span> k –<\/span> 1<\/span>)<\/span> +<\/span> MOD)<\/span> %<\/span> MOD<\/p>\n

                return<\/span> ans<\/p>\n

                if<\/span> __name__ ==<\/span> "__main__"<\/span>:<\/span>
                \n n =<\/span> 5<\/span>
                \n k =<\/span> 3<\/span>
                \n result =<\/span> count_valid_sequences(<\/span>n,<\/span> k)<\/span>
                \n print<\/span>(<\/span>result)<\/span><\/p>\n

                \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"<\/p>\n

                C++\u5b8c\u6574\u4ee3\u7801\u5982\u4e0b：<\/h2>\n

                #<\/span>include<\/span> <iostream><\/span><\/span>
                \nusing<\/span> namespace<\/span> std;<\/span><\/p>\n

                const<\/span> int<\/span> MOD =<\/span> 1'000'000'007<\/span>;<\/span>
                \nconst<\/span> int<\/span> MX =<\/span> 500'000<\/span>;<\/span><\/p>\n

                int<\/span> fac[<\/span>MX]<\/span>;<\/span> \/\/ fac[i] = i!<\/span>
                \nint<\/span> invF[<\/span>MX]<\/span>;<\/span> \/\/ invF[i] = i!^(-1) = pow(i!, MOD-2)<\/span><\/p>\n

                int<\/span> powmod<\/span>(<\/span>int<\/span> x,<\/span> int<\/span> n)<\/span> {<\/span>
                \n long<\/span> long<\/span> res =<\/span> 1<\/span>;<\/span>
                \n long<\/span> long<\/span> base =<\/span> x;<\/span>
                \n while<\/span> (<\/span>n ><\/span> 0<\/span>)<\/span> {<\/span>
                \n if<\/span> (<\/span>n &<\/span> 1<\/span>)<\/span> res =<\/span> res *<\/span> base %<\/span> MOD;<\/span>
                \n base =<\/span> base *<\/span> base %<\/span> MOD;<\/span>
                \n n >>=<\/span> 1<\/span>;<\/span>
                \n }<\/span>
                \n return<\/span> (<\/span>int<\/span>)<\/span>res;<\/span>
                \n}<\/span><\/p>\n

                void<\/span> init<\/span>(<\/span>)<\/span> {<\/span>
                \n fac[<\/span>0<\/span>]<\/span> =<\/span> 1<\/span>;<\/span>
                \n for<\/span> (<\/span>int<\/span> i =<\/span> 1<\/span>;<\/span> i <<\/span> MX;<\/span> ++<\/span>i)<\/span> {<\/span>
                \n fac[<\/span>i]<\/span> =<\/span> (<\/span>long<\/span> long<\/span>)<\/span>fac[<\/span>i –<\/span> 1<\/span>]<\/span> *<\/span> i %<\/span> MOD;<\/span>
                \n }<\/span><\/p>\n

                invF[<\/span>MX –<\/span> 1<\/span>]<\/span> =<\/span> powmod<\/span>(<\/span>fac[<\/span>MX –<\/span> 1<\/span>]<\/span>,<\/span> MOD –<\/span> 2<\/span>)<\/span>;<\/span>
                \n for<\/span> (<\/span>int<\/span> i =<\/span> MX –<\/span> 1<\/span>;<\/span> i ><\/span> 0<\/span>;<\/span> —<\/span>i)<\/span> {<\/span>
                \n invF[<\/span>i –<\/span> 1<\/span>]<\/span> =<\/span> (<\/span>long<\/span> long<\/span>)<\/span>invF[<\/span>i]<\/span> *<\/span> i %<\/span> MOD;<\/span>
                \n }<\/span>
                \n}<\/span><\/p>\n

                \/\/ \u4ece n \u4e2a\u6570\u4e2d\u9009 m \u4e2a\u6570\u7684\u65b9\u6848\u6570<\/span>
                \nint<\/span> comb<\/span>(<\/span>int<\/span> n,<\/span> int<\/span> m)<\/span> {<\/span>
                \n if<\/span> (<\/span>m <<\/span> 0<\/span> ||<\/span> m ><\/span> n)<\/span> return<\/span> 0<\/span>;<\/span>
                \n return<\/span> (<\/span>long<\/span> long<\/span>)<\/span>fac[<\/span>n]<\/span> *<\/span> invF[<\/span>m]<\/span> %<\/span> MOD *<\/span> invF[<\/span>n –<\/span> m]<\/span> %<\/span> MOD;<\/span>
                \n}<\/span><\/p>\n

                int<\/span> countValidSequences<\/span>(<\/span>int<\/span> n,<\/span> int<\/span> k)<\/span> {<\/span>
                \n int<\/span> ans =<\/span> comb<\/span>(<\/span>n –<\/span> 1<\/span>,<\/span> k –<\/span> 1<\/span>)<\/span>;<\/span>
                \n if<\/span> (<\/span>(<\/span>n +<\/span> k)<\/span> %<\/span> 2<\/span> ==<\/span> 0<\/span>)<\/span> {<\/span>
                \n ans =<\/span> (<\/span>ans –<\/span> comb<\/span>(<\/span>(<\/span>n +<\/span> k)<\/span> \/<\/span> 2<\/span> –<\/span> 1<\/span>,<\/span> k –<\/span> 1<\/span>)<\/span> +<\/span> MOD)<\/span> %<\/span> MOD;<\/span> \/\/ +MOD \u4fdd\u8bc1\u7b54\u6848\u975e\u8d1f<\/span>
                \n }<\/span>
                \n return<\/span> ans;<\/span>
                \n}<\/span><\/p>\n

                int<\/span> main<\/span>(<\/span>)<\/span> {<\/span>
                \n init<\/span>(<\/span>)<\/span>;<\/span><\/p>\n

                int<\/span> n =<\/span> 5<\/span>;<\/span>
                \n int<\/span> k =<\/span> 3<\/span>;<\/span>
                \n int<\/span> result =<\/span> countValidSequences<\/span>(<\/span>n,<\/span> k)<\/span>;<\/span>
                \n cout <<<\/span> result <<<\/span> endl;<\/span><\/p>\n

                return<\/span> 0<\/span>;<\/span>
                \n}<\/span><\/p>\n

                \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"<\/p>\n","protected":false},"excerpt":{"rendered":"

                2026-10-25：\u7edf\u8ba1\u6709\u6548\u5e8f\u5217\u6570\u76ee\u3002\u7528go\u8bed\u8a00，\u7ed9\u5b9a\u4e24\u4e2a\u6b63\u6574\u6570 n \u548c k\u3002\u73b0\u5728\u8981\u7edf\u8ba1\u6ee1\u8db3\u4e0b\u9762\u6761\u4ef6\u7684\u6709\u5e8f\u5e8f\u5217\u6570\u91cf：\u5e8f\u5217\u4e00\u5171\u5305\u542b k \u4e2a\u6b63\u6574\u6570，\u8fd9\u4e9b\u6570\u7684\u603b\u548c\u6070\u597d\u4e3a n，\u5e76\u4e14\u8fd9\u4e9b\u6570\u7684\u4e58\u79ef\u5fc5\u987b\u662f\u5076\u6570\u3002\u5e8f\u5217\u6309\u4f4d\u7f6e\u533a\u5206，\u53ea\u8981\u67d0\u4e2a\u4f4d\u7f6e\u4e0a\u7684\u6570\u4e0d\u540c，\u5c31\u7b97\u4f5c\u4e0d\u540c\u5e8f\u5217\u3002\u6700\u7ec8\u7ed3\u679c\u9700\u8981\u5bf9 1000000007 \u53d6\u4f59\u540e\u8fd4\u56de\u3002
                \n1 < n < 500000\u3002
                \n1 < k < n\u3002
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