组合数学

卢卡斯定理

求组合数:

模板题:P3807 【模板】卢卡斯定理:https://www.luogu.com.cn/problem/P3807

#include<bits/stdc++.h>
#define INF 0x3f3f3f3f
#define DOF 0x7f7f7f7f
#define endl ‘\n‘
#define mem(a,b) memset(a,b,sizeof(a))
#define debug(case,x); cout<<case<<"  : "<<x<<endl;
#define open freopen("ii.txt","r",stdin)
#define close freopen("oo.txt","w",stdout)
#define IO ios::sync_with_stdio(false);cin.tie(0);cout.tie(0)
#define pb push_back
using namespace std;
typedef long long ll;
typedef pair<int, int> pii;
typedef pair<long long, long long> PII;
const int maxn = 1e6 + 10;
const ll mod=1e9+7;

ll j[maxn];
ll n,m,p;
ll quick_pow(ll a,ll b ,ll p){
    ll ans = 1;
    while(b){
        if(b & 1) ans = (ans * a)%p;
        a = a * a % p;
        b>>=1;
    }
    return ans;
}
ll C(ll n,ll m){
    if(m > n)return 0;
    return ((j[n] * quick_pow(j[m],p-2,p))%p * quick_pow(j[n-m],p-2,p)%p);
}
ll Lucas(ll n,ll m){
    if(!m)return 1;
    return C(n%p,m%p)*Lucas(n/p,m/p)%p;
}
int main(){
    int T;
    j[0] = 1;
    cin>>T;
    while(T--){
        cin >> n >> m >> p;
        for(int i=1;i<=p;i++)
            j[i] = j[i-1] * i % p;
        cout<<Lucas(n+m,n)<<endl;
    }
    return 0;
}

链接:https://ac.nowcoder.com/acm/problem/16596
来源:牛客网

题目描述

给定一个多项式(ax+by)k,请求出多项式展开后xnym项的系数。

输入描述:

共一行,包含5个整数,分别为a,b,k,n,m,每两个整数之间用一个空格隔开。

输出描述:

输出共1行,包含一个整数,表示所求的系数,这个系数可能很大,输出对10007取模后的结果。

思路

\[T_{r+1}=C_{k}^{n}*(ax)^n*(by)^m\]

对公式直接求解

#include<bits/stdc++.h>
#define INF 0x3f3f3f3f
#define DOF 0x7f7f7f7f
#define endl ‘\n‘
#define mem(a,b) memset(a,b,sizeof(a))
#define debug(case,x); cout<<case<<"  : "<<x<<endl;
#define open freopen("ii.txt","r",stdin)
#define close freopen("oo.txt","w",stdout)
#define IO ios::sync_with_stdio(false);cin.tie(0);cout.tie(0)
#define pb push_back
using namespace std;
typedef long long ll;
typedef pair<int, int> pii;
typedef pair<long long, long long> PII;
const int maxn = 1e6 + 10;
const ll mod=1e9+7;

ll j[maxn];
ll n,m,p;
ll quick_pow(ll a,ll b ,ll p){
    ll ans = 1;
    while(b){
        if(b & 1) ans = (ans * a)%p;
        a = a * a % p;
        b>>=1;
    }
    return ans;
}
ll C(ll n,ll m){
    if(m > n)return 0;
    return ((j[n] * quick_pow(j[m],p-2,p))%p * quick_pow(j[n-m],p-2,p)%p);
}
ll Lucas(ll n,ll m){
    if(!m)return 1;
    return C(n%p,m%p)*Lucas(n/p,m/p)%p;
}


int main(){
    j[0]=1;
    ll a,b,k,n1,m1;cin>>a>>b>>k>>n1>>m1;
    n=k,m=n1;
    p=10007;
    for(int i=1;i<=p;++i){
        j[i]=j[i-1]*i%p;
    }
    cout<<(((quick_pow(a,n1,p)%p*(quick_pow(b,m1,p)%p))%p)*(Lucas(n,m)%p))%p<<endl;
}

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