UVa 10375 Choose and divide (组合数相除)

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本文探讨如何高效计算并比较复杂组合数的值，通过实例解析算法优化过程，实现快速准确的结果输出。

10375 - Choose and divide

Time limit: 3.000 seconds

http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=115&page=show_problem&problem=1316

The binomial coefficient C(m,n) is defined as

         m!
C(m,n) = --------
         n!(m-n)!

Given four natural numbers p, q, r, and s, compute the the result of dividing C(p,q) by C(r,s).

The Input

Input consists of a sequence of lines. Each line contains four non-negative integer numbers giving values for p, q, r, and s, respectively, separated by a single space. All the numbers will be smaller than 10,000 with p>=q and r>=s.

The Output

For each line of input, print a single line containing a real number with 5 digits of precision in the fraction, giving the number as described above. You may assume the result is not greater than 100,000,000.

Sample Input

10 5 14 9
93 45 84 59
145 95 143 92
995 487 996 488
2000 1000 1999 999
9998 4999 9996 4998

Output for Sample Input

0.12587
505606.46055
1.28223
0.48996
2.00000
3.99960

思路：不要被题目那个公式误导了~自己算一算几个组合数，比如C(100,4)，发现分子分母都是4个乘数，那就好办了，写一个for循环每次一除一乘就是。

完整代码：

/*0.022s*/

#include<cstdio>
#include<algorithm>
using namespace std;

int main()
{
	int p, q, r, s,maxn,i;
	double ans;
	while (~scanf("%d%d%d%d", &p, &q, &r, &s))
	{
		q = max(q, p - q), s = max(s, r - s);
		maxn = max(q, s);
		ans = 1.0;
		for (i = 1; i <= maxn; ++i)
		{
			if (i <= q) ans = ans / i * (p - q + i);
			if (i <= s) ans = ans / (r - s + i) * i;
		}
		printf("%.5f\n", ans);
	}
	return 0;
}