POJ 3219 Binomial Coefficients

思想深奥，代码操简单的数学问题。

题目链接：http://poj.org/problem?id=3219

CSUST 2012年暑假8月组队后个人赛第12场：http://acm.hust.edu.cn:8080/judge/contest/view.action?cid=11900#problem/D

D - Binomial Coefficients

Time Limit:1000MS     Memory Limit:131072KB     64bit IO Format:%I64d & %I64u

Submit  Status  Practice  POJ 3219

Description

The binomial coefficient C(n, k) has been extensively studied for its importance in combinatorics. Binomial coefficients can be recursively defined as follows:

C(n, 0) = C(n, n) = 1 for all n > 0;
C(n, k) = C(n − 1, k − 1) + C(n − 1, k) for all 0 < k < n.

Given n and k, you are to determine the parity of C(n, k).

Input

The input contains multiple test cases. Each test case consists of a pair of integers n and k (0 ≤ k ≤ n < 2³¹, n > 0) on a separate line.

End of file (EOF) indicates the end of input.

Output

For each test case, output one line containing either a “0” or a “1”, which is the remainder of C(n, k) divided by two.

Sample Input

1 1
1 0
2 1

Sample Output

1
1
0

题意：就是判断C（n,k）的奇偶性，如果为奇数就输出1否则输出0.

思路：按照杨辉三角找规律即可。C(n,k)(k<=n)的奇偶性取决于(n-k)与k的二进制表达式是否存在同一位上的两个数码均为1，若存在，则为偶数，反之为奇数。

证明： http://file.lw23.com/2/2a/2a6/2a623be7-8928-456d-82b5-d9818f808555.pdf

代码：

#include<stdio.h>
int n,k;//对于这个程序，定义全局变量比较快0ms过，否则要16ms
int main()
{
    while(scanf("%d%d",&n,&k)!=EOF)
    {
        printf("%d\n",k&(n-k) ? 0 : 1);
    }
    return 0;
}