Unknown Treasure
Time Limit: 1500/1000 MS (Java/Others) Memory Limit: 131072/131072 K (Java/Others)
Total Submission(s): 2209 Accepted Submission(s): 821
Problem Description
On the way to the next secret treasure hiding place, the mathematician discovered a cave unknown to the map. The mathematician entered the cave because it is there. Somewhere deep in the cave, she found a treasure chest with a combination lock and some numbers on it. After quite a research, the mathematician found out that the correct combination to the lock would be obtained by calculating how many ways are there to pick m different apples among n of them and modulo it with M. M is the product of several different primes.
Input
On the first line there is an integer T(T≤20) representing the number of test cases.
Each test case starts with three integers n,m,k(1≤m≤n≤1018,1≤k≤10) on a line where k is the number of primes. Following on the next line are k different primes p1,...,pk. It is guaranteed that M=p1⋅p2⋅⋅⋅pk≤1018 and pi≤105 for every i∈{1,...,k}.
Output
For each test case output the correct combination on a line.
Sample Input
1
9 5 2
3 5
9 5 2
3 5
Sample Output
6
Source
Recommend
分析：
根据Lucas求解中 每一个i：C(n,m)%pi,然后根据中国剩余定理把这些结果整合起来。就能得到答案。
注意中国剩余定理在计算当前余数与其他余数乘积的最小公倍数的gcd为1的值时候，由于数据量太大，要取模。
#include<iostream>
#include<stdio.h>
using namespace std;
long long pri[];
long long a[];
long long ext_gcd(long long a,long long b,long long *x,long long *y)
{
if(b==)
{
*x=,*y=;
return a;
}
long long r = ext_gcd(b,a%b,x,y);
long long t = *x;
*x= *y;
*y = t - a/b * *y;
return r;
}
long long quick_mod(long long n,long long m,long long mod)
{
long long ans=;
while(m)
{
if(m&)
ans=(ans*n)%mod;
m>>=;
n=(n*n)%mod;
}
return ans%mod;
}
long long get_c(long long n,long long m,long long mod)
{
long long a=,b=;
for(int i=; i<=m; i++)
{
b=b*i%mod;
a=a*(n-i+)%mod;
}
return (a*(quick_mod(b,mod-,mod)))%mod;
}
long long Lucas(long long n,long long m,long long mod)
{
if(m==) return ;
return (Lucas(n/mod,m/mod,mod)*get_c(n%mod,m%mod,mod))%mod;
}
long long mul(long long a,long long n,long long mod)
{
a = (a%mod+mod)%mod;
n = (n%mod+mod)%mod;
long long ret =;
while(n)
{
if(n&)
ret=(ret+a)%mod;
a=(a+a)%mod;
n>>=;
}
return ret%mod;
}
long long chinese_reminder(long long a[],long long pri[],int len)
{
long long mul_pri=;
long long res=;
for(int i=;i<len;i++)
{
mul_pri*=pri[i];
}
for(int i=;i<len;i++)
{
long long m = mul_pri/pri[i];
long long x,y;
ext_gcd(pri[i],m,&x,&y);
//res=(res+y*m*a[i])%mul_pri;
res=(res+mul(mul(y,m,mul_pri),a[i],mul_pri))%mul_pri;
}
return ((res%mul_pri+mul_pri)%mul_pri);
}
int main()
{
int t;
long long n,m;
int k;
scanf("%d",&t);
while(t--)
{
scanf("%I64d%I64d%d",&n,&m,&k);
for(int i=; i<k; i++)
scanf("%I64d",pri+i);
for(int i=; i<k; i++)
{
a[i]=Lucas(n,m,pri[i]);
}
long long ans = chinese_reminder(a,pri,k);
printf("%I64d\n",ans);
}
return ;
}