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: Time Limit Exceeded)

(defmacro reads* (syms &body body)
  `(let* (,@(mapcar #'(lambda (sym)
			(if (listp sym) sym `(,sym (read))))
		    syms))
     ,@body))
			    
  
(reads* (n a (lst (loop repeat n collect (read))) (sum (reduce #'+ lst)))
  (let ((count 0))
    (labels ((back-tracking (depth sum lst ave)
	       (cond ((zerop depth) (when (= sum ave) (incf count)))
		     ((<= ave sum) nil)
		     ((null lst) nil)
		     (t (back-tracking (1- depth)
				       (+ sum (car lst))
				       (cdr lst)
				       ave)
			(back-tracking depth
				       sum
				       (cdr lst)
				       ave)))))
      (loop
	 for lim from a to sum by a
	 finally (princ count)
	 do (back-tracking (/ lim a) 0 lst lim)))))