NCR和反阶乘(MODM）的实现

implementation of nCr and inverse factorial (MODm) for very large numbers

本文关键字：实现 MODM 阶乘 NCR      更新时间：2023-10-16

嗨，我在代码sprint5问题中实现NCR MODM时有问题。链接到问题是......https://www.hackerrank.com/contests/codesprint5/challenges/matrix-tracing。我还学到的是，我可以将泥泞算术规则应用于阶乘计算和反阶乘计算，以及计算功率（a，b）modmmm。但我不知道我缺少什么导致错误的答案。这是我当前的代码。

#include <cmath>
#include <cstdio>
#include <vector>
#include <iostream>
#include <algorithm>
#include <map>
#include<math.h>
using namespace std;
const int md = 1000000007;
const int co = 2000020;
unsigned long long int ft[co];
long long int fact(unsigned long long int n)
{   
   return ft[n];
}
void fct(){
    ft[1]=1;
    for(unsigned long long int i = 2;i<=2000020;i++){
        ft[i]=(i*ft[i-1]) % md;
        }
    }
long long int pow(long long int x, long long int n, long long int mod){
    long long int result=1; 
    while(n>0){
        if(n%2 ==1){
            result = (result*x) % mod;
        }
        n= n>>1;
        x= (x*x)% mod;  
    }
    return result;
}
int main() {
    /* Enter your code here. Read input from STDIN. Print output to STDOUT */  
    unsigned long long int m , n;
    long long result;
    int T;
    fct();
    cin>>T;
    while(T--){
        cin>>m>>n; 
        unsigned long long int mod = md-2;
         result = (fact(m+n-2) * pow( ( fact(m-1) * fact(n-1) ) , mod, md )) % md ;
        cout<<result<<endl;
    }
    return 0;
}

最后我在代码中得到了错误。

错误....

  1. 我应该使用常数变量md和co作为无签名的长长长。int而不是int
  2. 第二个错误是用于计算pow(a,b) % md .....在pow()中的算法中的错误功能，我应该先进行x % md，然后再进行进一步处理因为X可以通过大于md的概率。

当前的工作代码是..... 

#include <cmath>
#include <cstdio>
#include <vector>
#include <iostream>
#include <algorithm>
#include <map>
#include<math.h>
using namespace std;
const unsigned long long int md = 1000000007; 
const unsigned long long int co = 2000020;
unsigned long long int ft[co];
unsigned long long int fact(unsigned long long int n)
{   
    return ft[n];
}
void fct(){
    ft[0]=1;
    for(unsigned long long int i = 1;i<=2000020;i++){
        ft[i]=(i*ft[i-1]) % md;
    }
}
unsigned long long int pow(unsigned long long int x, unsigned long long int n, unsigned long long int mod){
    unsigned long long int result=1; 
    x = x % md;
    while(n>0){
        if(n%2 ==1){
            result = (result*x) % md;
        }
        n= n>>1;
        x= (x*x)% md;   
    }
    return result;
}
int main() {
    /* Enter your code here. Read input from STDIN. Print output to STDOUT */  
    unsigned long long int m , n;
    unsigned long long int result;
    int T;
    fct();
    cin>>T;
    while(T--){
        cin>>m>>n; 
        unsigned long long int mod = md-2;   
        result = (fact(m+n-2) * pow( ( fact(m-1) * fact(n-1) ) , mod, md )) % md ;
        cout<<result<<endl; 
    }
    return 0;
}
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