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 Common Lisp (SBCL 1.1.14) (judge verdict: Accepted)

(let* ((n (read))
			 (a (read))
			 (x (make-array n :initial-contents (loop :repeat n :collect (read))))
			 (cache (make-array (expt 51 4) :initial-element nil)))
	(defun s (v k i)
		(cond ((= i n) (if (and (> v 0) (= v (* k a))) 1 0))
					((> v (* a n)) 0)
					(t (let ((key (+ (* v 51 51) (* k 51) i)))
							 (if (aref cache key)
									 (aref cache key)
									 (setf (aref cache key) (+ (s v k (1+ i))
																						 (s (+ v (aref x i)) (1+ k) (1+ i)))))))))
  (format t "~a~%" (s 0 0 0)))