A. Kyoya and Colored Balls_排列组合，组合数

最新推荐文章于 2021-07-21 14:27:51 发布

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最新推荐文章于 2021-07-21 14:27:51 发布

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原文链接：http://www.cnblogs.com/TreeDream/p/5346542.html

本文解析了Codeforces Round #309 (Div.1)中A题「彩色球的排列方式」的解题思路，包括从后往前思考的方法、递推公式以及组合数的使用，详细阐述了如何通过组合数学解决此类问题。

Codeforces Round #309 (Div. 1)

Kyoya Ootori has a bag with n colored balls that are colored with k different colors. The colors are labeled from 1 to k. Balls of the same color are indistinguishable. He draws balls from the bag one by one until the bag is empty. He noticed that he drew the last ball of color ibefore drawing the last ball of color i + 1 for all i from 1 to k - 1. Now he wonders how many different ways this can happen.

Input

The first line of input will have one integer k (1 ≤ k ≤ 1000) the number of colors.

Then, k lines will follow. The i-th line will contain ci, the number of balls of the i-th color (1 ≤ ci ≤ 1000).

The total number of balls doesn't exceed 1000.

Output

A single integer, the number of ways that Kyoya can draw the balls from the bag as described in the statement, modulo 1 000 000 007.

Examples

input

output

input

output

Note

In the first sample, we have 2 balls of color 1, 2 balls of color 2, and 1 ball of color 3. The three ways for Kyoya are:


1 2 1 2 3
1 1 2 2 3
2 1 1 2 3

解题报告：

1、可以从后往前思考，先把第n种颜色的，最后一个球放到最后，然后将这个颜色的其余的球随便放，然后将第(n-1)种颜色的球放到，之前放的球的最前一个的前面，递推下去。

2、递推公式：

for(int i=n;i>=1;i--)
{
    if(cnt[i]==0) continue;
    ans=(ans*C[c-1][cnt[i]-])% mod;
    c-=cnt[i];
}

3、组合数递推公式：

void init()
{
    memset(C, 0, sizeof(C));
    C[0][0] = 1;
    C[1][0] = C[1][1] = 1;
    for(int i = 2; i <= 1000; ++i)
    {
        C[i][0] = C[i][i] = 1;
        for(int j = 1; j < i; ++j)
        {
            C[i][j] = (C[i-1][j-1] + C[i-1][j]) % mod;
        }
    }
}

#include <bits/stdc++.h>

using namespace std;
typedef long long ll;

const int maxn = 1010;
const ll  mod  = 1000000007;
ll C[maxn][maxn];

void init()
{
    memset(C, 0, sizeof(C));
    C[0][0] = 1;
    C[1][0] = C[1][1] = 1;
    for(int i = 2; i <= 1000; ++i)
    {
        C[i][0] = C[i][i] = 1;
        for(int j = 1; j < i; ++j)
        {
            C[i][j] = (C[i-1][j-1] + C[i-1][j]) % mod;
        }
    }
}

int cnt[maxn];

int main()
{

    init();
    int n;
    int c = 0;
    ll  ans = 1;
    scanf("%d", &n);
    for(int i = 1; i <= n; i++)
    {
        scanf("%d", &cnt[i]);
        c += cnt[i];
    }

    for(int i = n; i >= 1; i--)
    {
        if(cnt[i] == 0) continue;
        ans = (ans * C[c-1][cnt[i]-1]) % mod;
        c -= cnt[i];
    }

    cout << ans << endl;

    return 0;
}