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 OCaml (4.02.3) (judge verdict: Time Limit Exceeded)

let get_line () =
    read_line ()
    |> Str.split (Str.regexp " ")

(* use List.cons in 4.03.0 or later *)
let cons elem lst = elem :: lst

let get_sublists lst =
    let rec loop lst acc =
        match lst with
        | [] -> acc
        | x :: xs ->
            loop xs 
                (List.append (List.map (cons x) acc) acc)
    in
    loop lst [[]]

let get_average lst =
    let sum = List.fold_left ( +. ) 0.0 lst in
    sum /. float_of_int (List.length lst)

let print_list lst =
    List.map string_of_float lst
    |> String.concat " "
    |> print_endline

let () =
    let a =
        let [_; a] = get_line () in
        float_of_string a
    in
    let cards =
        get_line ()
        |> List.map float_of_string
    in
    get_sublists cards
    |> List.map get_average
    |> List.filter (fun x -> x = a)
    |> List.length
    |> print_int
    |> print_newline