Problem:
Score : 300 points

Problem Statement

Tak has N cards. On the i-th (1 \leq i \leq N) card is written an integer x_i.
He is selecting one or more cards from these N cards, so that the average of the integers written on the selected cards is exactly A.
In how many ways can he make his selection?

Constraints

1 \leq N \leq 50

1 \leq A \leq 50

1 \leq x_i \leq 50

N,\,A,\,x_i are integers.

Partial Score

200 points will be awarded for passing the test set satisfying 1 \leq N \leq 16.

Input

The input is given from Standard Input in the following format:

N A
x_1 x_2 ... x_N

Output

Print the number of ways to select cards such that the average of the written integers is exactly A.

Sample Input 1

4 8
7 9 8 9

Sample Output 1

5

The following are the 5 ways to select cards such that the average is 8:

Select the 3-rd card.

Select the 1-st and 2-nd cards.

Select the 1-st and 4-th cards.

Select the 1-st, 2-nd and 3-rd cards.

Select the 1-st, 3-rd and 4-th cards.

Sample Input 2

3 8
6 6 9

Sample Output 2

0

Sample Input 3

8 5
3 6 2 8 7 6 5 9

Sample Output 3

19

Sample Input 4

33 3
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

Sample Output 4

8589934591

The answer may not fit into a 32-bit integer.

Solution in Python (3.4.3) (judge verdict: Runtime Error)

from collections import Counter

from operator import mul
from functools import reduce
def cmb(n,r):
	if n < r:return 0
	if r == 0 or n-r == 0:return 1
	r = min(n-r,r)
	u = reduce(mul, range(n, n-r, -1))
	d = reduce(mul, range(1,r+1))
	return u//d

n,a,*x = map(int,open(0).read().split())
dp = [[0]*(n*max(x)+1)for _ in range(n+1)]
dp[0][0] = 1
lim1 = lim2 = 1
for k,v in Counter(x).items():
	for i in range(1,v+1):
		c = cmb(v,i)
		for j in range(lim1)[::-1]:
			for s in range(lim2)[::-1]:
				dp[j+i][s+k*i] += dp[j][s]*c
	lim1 += v
	lim2 += k*v
cnt = 0
for i in range(1,n+1):
	cnt += dp[i][i*a]
print(cnt)