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 (2.7.6) (judge verdict: Time Limit Exceeded)

#!/usr/bin/env python
N, A = map(int, raw_input().split())
x = map(int, raw_input().split())

x1 = x[:N // 2]
x2 = x[N // 2:]
hoge = [(0, 0)]

for num in x1:
    sz = len(hoge)
    for i in range(sz):
        p = (hoge[i][0] + 1, hoge[i][1] + num)
        hoge.append(p)

dic = {}
for p in hoge:
    rem = A * p[0] - p[1]
    if not rem in dic:
        dic[rem] = 1
    else:
        dic[rem] += 1

hoge = [(0, 0)]

for num in x2:
    sz = len(hoge)
    for i in range(sz):
        p = (hoge[i][0] + 1, hoge[i][1] + num)
        hoge.append(p)

ans = 0
for p in hoge:
    rem = p[1] - A * p[0]
    if rem in dic:
        ans += dic[rem]

print ans - 1