JOIN

 Problem Statement

Problem Statement for Permutant

### Problem Statement

Hero likes combinatorics. He is especially fond of complicated formulas related to permutations. Hero has a int[] a containing n (not necessarily distinct) positive integers. The sum of these integers is m.

There are n! permutations of the array a. For each of these permutations, Hero does the following computation:

1. Let b be the permuted array a.
2. Let s be the array of prefix sums of the array b. That is, for each i, s[i] is the sum of the first i elements of b.
3. Hero computes the value m! / (s[2] * s[3] * ... * s[n]). Note that the product in the denominator starts with s[2], not s[1].

For example, suppose that b = {3,1,2}. This means that m = 3+1+2 = 6. The relevant prefix sums are s[2] = 3+1 = 4 and s[3] = 3+1+2 = 6. The value Hero computes is 6! / (4 * 6) = 30.

At the end, Hero will have n! values (one for each possible permutation of a). Let X be the sum of all those values. Help him by computing and returning the value (X modulo 1,000,000,007).

### Definition

 Class: Permutant Method: counthis Parameters: int[] Returns: int Method signature: int counthis(int[] a) (be sure your method is public)

### Notes

-It can easily be shown that each of the n! values computed by Hero is a positive integer.

### Constraints

-Number of elements in a will be between 1 and 50, inclusive.
-Each element in a will be between 1 and 50, inclusive.
-Sum of elements in a will be between 1 and 1000, inclusive.

### Examples

0)

 `{1,1}`
`Returns: 2`
 For the given a we have n = 2 and m = 1+1 = 2. There are n! = 2 permutations of a. For each of them, we have b = {1,1}, s[2] = 2, and thus Hero computes the value 2! / 2 = 1. The sum of all computed values is X = 1 + 1 = 2.
1)

 `{1,2}`
`Returns: 4`
2)

 `{1,2,3}`
`Returns: 188`
 The six permutations of a give us the following six values computed by Hero: 24, 24, 30, 30, 40, and 40.
3)

 `{1,1,1,1,1}`
`Returns: 120`
4)

 `{1,2,2,3,3,3,4,4,4,4}`
`Returns: 860993751`

This problem statement is the exclusive and proprietary property of TopCoder, Inc. Any unauthorized use or reproduction of this information without the prior written consent of TopCoder, Inc. is strictly prohibited. (c)2010, TopCoder, Inc. All rights reserved.

This problem was used for:
Single Round Match 684 Round 1 - Division I, Level Three