poj 2249 Binomial Showdown(组合数 公式优化)

//  组合数学,开始了……

题目地址 : poj 2249 Binomial Showdown

Description

In how many ways can you choose k elements out of n elements, not taking order into account? 
Write a program to compute this number.

Input

The input will contain one or more test cases. 
Each test case consists of one line containing two integers n (n>=1) and k (0<=k<=n). 
Input is terminated by two zeroes for n and k.

Output

For each test case, print one line containing the required number. This number will always fit into an integer, i.e. it will be less than 2³¹. 
Warning: Don't underestimate the problem. The result will fit into an integer - but if all intermediate results arising during the computation will also fit into an integer depends on your algorithm. The test cases will go to the limit. 

Sample Input

4 210 549 60 0

Sample Output

625213983816

Source

Ulm Local 1997

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组合数公式的优化

普通的组合数公式：C(n,m) = n!/((n-m)!*m!) = (n * n-1 * n-2 * n-3 * n-4 * n-5 * n-6 * …… * n-m) / (m * m-1 * m-2 * m-3 * m-4 * m-5 * …… * 2 * 1)

优化 ：将分子和分母 的因子分别存到一个数组中 nn[]  (分子)  mm[] (分母),双重循环遍历, 进行各个因子约分，因为 组合数为一个整数 ，即N % M = 0,

所以mm[] 的元素一定可以全部约分为1，然后只需将 mm[] 中的元素相乘就行了。。

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#include <iostream>#include<string.h>#include <stdio.h>using namespace std;const int N = 10000;int nn[N],mm[N];int gcd(int a,int b)//  求n,m 的最大公约数{    return (b==0)?a:gcd(b,a%b);}int main(){    int i,j,n,m,t,h,sum,temp;    while(cin>>n>>m&&(n||m))//  程序结束条件是 n,m 中一个为0就行，用||,我开始用&& WA了好多次，要注意    {        t = h = 0;        sum = 1;        //cout<<gcd(m,n)<<endl;        if(m>n-m)            m = n-m;        for(i = n-m+1;i<=n;i++)// 分子赋值            nn[t++] = i;        for(i = 1;i<=m;i++)//  分母赋值            mm[h++] = i;        for(i = 0;i<m;i++)        {            for(j = 0;j<m;j++)            {                temp = gcd(nn[i],mm[j]);//  约分                nn[i] = nn[i]/temp;                mm[j] = mm[j]/temp;            }        }        for(i = 0;i<m;i++)            sum*=nn[i];        cout<<sum<<endl;    }}