Problem:
Score : 800 points

Problem Statement

Given are N pairwise distinct non-negative integers A_1,A_2,\ldots,A_N.
Find the number of ways to choose a set of between 1 and K numbers (inclusive) from the given numbers so that the following two conditions are satisfied:

The bitwise AND of the chosen numbers is S.

The bitwise OR of the chosen numbers is T.

Constraints

1 \leq N \leq 50

1 \leq K \leq N

0 \leq A_i < 2^{18}

0 \leq S < 2^{18}

0 \leq T < 2^{18}

A_i \neq A_j (1 \leq i < j \leq N)

Input

Input is given from Standard Input in the following format:

N K S T
A_1 A_2 ... A_N

Output

Print the answer.

Sample Input 1

3 3 0 3
1 2 3

Sample Output 1

2

The conditions are satisfied when we choose \{1,2\} or \{1,2,3\}.

Sample Input 2

5 3 1 7
3 4 9 1 5

Sample Output 2

2

Sample Input 3

5 4 0 15
3 4 9 1 5

Sample Output 3

3

Solution in Python (3.8.2) (judge verdict: Time Limit Exceeded)

from functools import lru_cache

n, k, and_, or_ = map(int, input().split())
lst = sorted([int(n) for n in input().split()])
lst = [n for n in lst if n & and_ == and_ and n | or_ == or_]
n = len(lst)
# print(lst)


@lru_cache(None)
def fac(n):
    return 1 if n <= 1 else n * fac(n - 1)


def choose(a, b):
    return fac(a) // fac(b) // fac(a - b)


@lru_cache(None)
def f(i, k, a, b):
    if a == and_ and b == or_:
        return sum(choose(n - i, x) for x in range(1 + min(k, n - i)))
    elif i == n:
        return 0
    else:
        return (f(i + 1, k - 1, a & lst[i], b | lst[i]) if k > 0 else 0) + f(i + 1, k, a, b)


print(sum(f(i + 1, k - 1, x, x) for i, x in enumerate(lst)))