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: Time Limit Exceeded)

b = []
def binomial(n):
    global b
    b = [None] * n
    for i in range(n):
        b[i] = [0] * n
        b[i][i] = 1
    for i in range(1, n):
        for j in range(1, i + 1):
            b[i][j] = b[i - 1][j - 1] + b[i - 1][j]
    
cache = {}

def s(x, a, k, i):
    if a < 0: return 0
    if k < 0: return 0
    if k == 0:
        if a == 0: return 1
        else: return 0
    if i > a: return 0
    if i == 51: return 0
    key = (a, k, i)
    if key in cache: return cache[key]
    c = s(x, a, k, i + 1)
    for j in range(1, x[i] + 1):
        c += b[x[i] + 1][j + 1] * s(x, a - i * j, k - j, i + 1)
    cache[key] = c
    return c

n, a = map(int, input().split())
binomial(n + 2)
x = [0] * 51
for k in input().split():
    x[int(k)] += 1

c = 0
for i in range(1, n + 1):
    c += s(x, a * i, i, 0)
print(c)