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: Accepted)

#x枚目まででy枚選んで合計がzになる場合の数dp[x][y][z](x+1*y+1*たくさん)
N,A=map(int,input().split())
x=[int(i) for i in input().split()]
dp=[[[0]*(50*N+1) for i in range(N+1)]for k in range(N+1)]
ans=0
dp[0][0][0]=1
for made,i in enumerate(dp):#走査して、一個上を書き換え
    if made>N:break
    for maisu,j in enumerate(i):
        for wa,k in enumerate(j):
            if k and made<N:#made枚目までにmaisu枚選んでwaになる場合の数
                dp[made+1][maisu][wa]+=k#made+1枚目についての議論
                newwa=wa+x[made]
                if newwa<=A*N:
                    dp[made+1][maisu+1][newwa]+=k
for mai in range(1,N+1):
    a=dp[N][mai][mai*A]
    if a:
        ans+=a
print(ans)