HDU 3944 组合数学+数论

/********************************************************************************    很好的组合数学题，跟数论结合，Lucas定理的应用，关键是这个转化：C(n, k) % p = C(n % p, k % p) * C(n / p, k / p) % pC(n, k) % p = n! / (k! * (n - k)!) % p = n! * (k! * (n - k)!)^-1 % p现在要求(k! * (n - k)!) % p的逆元，因为k!*(n - k)!不可能包含素因子p，所以逆元必然存在。注意求得的逆元可能小于0！然后dfs下就完事，但是因为每次求阶乘显然会超时，所以事先打表处理～(PS:此段出处Racebug)切数论题还是要小心。。。整数域跟素数域都很容易坑爹的～********************************************************************************/#include <iostream>#include <algorithm>#include <cstdlib>#include <cstring>#include <utility>#include <cstdio>#include <memory>#include <string>#include <vector>#include <cmath>#include <ctime>#include <queue>#include <stack>#include <map>#include <set>using namespace std;//typedef long long LL;typedef __int64 LL;typedef pair<int, int> PII;typedef pair<double, double> PDD;typedef map<int, int>::iterator MI;typedef vector<int>::iterator VI;typedef set<int>::iterator SI;const int INF_INT = 0x3f3f3f3f;const double oo = 10e9;const double eps = 10e-7;const double PI = acos(-1.0);const int MAXN = 10004;int pcnt, prm[MAXN], dp[1500][MAXN];bool is[MAXN];inline bool scan_d(int &num){char in;in = getchar();if(EOF == in) {return false;}while(in < '0' || in > '9'){in = getchar();}num = in - '0';while(in = getchar(), in >= '0' && in <= '9'){num *= 10, num += in - '0';}return true;}void preprocess(){pcnt = 0;memset(is, true, sizeof(is));is[0] = is[1] = false;prm[pcnt++] = 2;for (int i = 4; i < MAXN; i += 2){is[i] = false;}int ind;for (ind = 3; ind * ind <= MAXN; ind += 2){if (is[ind]){prm[pcnt++] = ind;for (int j = ind * ind, k = ind * 2; j < MAXN; j += k){is[j] = false;}}}while (ind < MAXN){if (is[ind]){prm[pcnt++] = ind;}ind += 2;}for (int i = 0; i < pcnt; ++i){dp[i][0] = 1;for (int j = 1; j < MAXN; ++j){dp[i][j] = (dp[i][j - 1] * j) % prm[i];}}return ;}int ext_gcd(int a, int b, int &x, int &y){if (0 == b){x = 1;y = 0;return a;}int d = ext_gcd(b, a % b, x, y);int t = x;x = y;y = t - a / b * y;return d;}int bisearch(int p){int low = 0, high = pcnt - 1, mid = (low + high) >> 1;int res = mid;while (low <= high){mid = (low + high) >> 1;if (p == prm[mid]){res = mid;break ;}if (p < prm[mid]){high = mid - 1;}else{low = mid + 1;}}return res;}int Lucas_dfs(int n, int k, int id){int p = prm[id];int _n, _k, buf, tmp, _tmp, y;_n = n / p;_k = k / p;n %= p;k %= p;if (n < k || _n < _k){return 0;}tmp = (dp[id][k] * dp[id][n - k]) % p;ext_gcd(tmp, p, _tmp, y);if (_tmp < 0){_tmp += p;}buf = (dp[id][n] * _tmp) % p;if (0 == _n && 0 == _k){return buf;}return buf * Lucas_dfs(_n, _k, id);}void ace(){int cas = 0;int n, k, p;int id;while (scan_d(n)){scan_d(k);scan_d(p);if (k > n - k){k = n - k;}id = bisearch(p);printf("Case #%d: %d\n", ++cas, (Lucas_dfs(n + 1, k, id) + n - k) % p);}return ;}int main(){preprocess();ace();return 0;}