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 Perl (v5.18.2) (judge verdict: Accepted)

($N,$A)=glob<>;@y=map$_-$A,@x=(0,glob<>);$_>$X&&($X=$_)for@x,$A;
for$j(0..$N){for$t(0..2*$N*$X){
	$dp[$j][$t]=
	(
		  $j==0 &&  $t==$N*$X                            ? 1
		: $j>=1 && ($t-$y[$j]< 0 || $t-$y[$j]> 2*$N*$X ) ? $dp[$j-1][$t]
		: $j>=1 &&  0<=$t-$y[$j] && $t-$y[$j]<=2*$N*$X   ? $dp[$j-1][$t] + $dp[$j-1][$t-$y[$j]]
		:                                                  0
	)
}}
print$dp[$N][$N*$X]-1,$/