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: Accepted)

from collections import defaultdict

N,A = map(int, raw_input().split())
cards = map(int, raw_input().split())

class CardSet:
  def __init__(self, n):
    self.cards = []
    self.combis = [defaultdict(lambda:0) for i in range(n+1)]

  def add_card(self, card):
    self.cards.append(card)
    for n in range(len(self.cards)-1,0,-1):
      for k, v in self.combis[n].items():
        self.combis[n+1][k+card] += v
    self.combis[1][card] += 1

  def avg_count(self, average):
    count = 0
    for n in range(1,len(self.cards)+1):
      count += self.combis[n][n*average]
    return count

  def __str__(self):
    return '\n'.join([str(cs) for cs in self.combis])

cardset = CardSet(N)
for card in cards:
  cardset.add_card(card)
print cardset.avg_count(A)