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 PyPy3 (2.4.0) (judge verdict: Accepted)

n, a = map(int, input().split())
x = list(map(int, input().split()))
#dp[i][j][k] := i枚目までカードを選び、合計がjになり、k枚カードを選んだ選び方の総数
#合計のマックスは、xの総和で考える、高々2500なので間に合う
dp = [[[0 for i in range(n+1)] for j in range(sum(x)+1)] for k in range(n+1)]
dp[0][0][0] = 1
for i in range(1,n+1):
    for j in range(sum(x) + 1):
        for k in range(n+1):
            if j + x[i-1] < sum(x) + 1 and k < n:
                dp[i][j + x[i-1]][k+1] += dp[i-1][j][k]
            dp[i][j][k] += dp[i-1][j][k]
ans = 0
for j in range(sum(x)+1):
    for k in range(1,n+1):
        if j == a * k:
            ans += dp[-1][j][k]

print(ans)