Problem:
Score : 200 points

Problem Statement

There are N bags of biscuits. The i-th bag contains A_i biscuits.

Takaki will select some of these bags and eat all of the biscuits inside.
Here, it is also possible to select all or none of the bags.

He would like to select bags so that the total number of biscuits inside is congruent to P modulo 2.
How many such ways to select bags there are?

Constraints

1 \leq N \leq 50

P = 0 or 1

1 \leq A_i \leq 100

Input

Input is given from Standard Input in the following format:

N P
A_1 A_2 ... A_N

Output

Print the number of ways to select bags so that the total number of biscuits inside is congruent to P modulo 2.

Sample Input 1

2 0
1 3

Sample Output 1

2

There are two ways to select bags so that the total number of biscuits inside is congruent to 0 modulo 2:

Select neither bag. The total number of biscuits is 0.

Select both bags. The total number of biscuits is 4.

Sample Input 2

1 1
50

Sample Output 2

0

Sample Input 3

3 0
1 1 1

Sample Output 3

4

Two bags are distinguished even if they contain the same number of biscuits.

Sample Input 4

45 1
17 55 85 55 74 20 90 67 40 70 39 89 91 50 16 24 14 43 24 66 25 9 89 71 41 16 53 13 61 15 85 72 62 67 42 26 36 66 4 87 59 91 4 25 26

Sample Output 4

17592186044416

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.math.BigInteger;
import java.util.Scanner;

public class Main {
  public static void main(String[] args) {
    Scanner sc = new Scanner(System.in);
    int n = sc.nextInt();
    int p = sc.nextInt();
    int odd = 0;
    for (int i = 0; i < n; i++) {
      if (sc.nextInt() % 2 == 1) {
        odd++;
      }
    }
    int even = n - odd;
//    System.out.println("o: " + odd);
//    System.out.println("e: " + even);
    BigInteger res;
    BigInteger en = BigInteger.ZERO;
    BigInteger on = BigInteger.ZERO;
    for (int i = 0; i <= even; i++) {
      en = en.add(choose(BigInteger.valueOf(even), BigInteger.valueOf(i)));
    }
    if (p == 0) {
      for (int i = 0; i <= odd; i += 2) {
        on = on.add(choose(new BigInteger("" + odd), new BigInteger("" + i)));
      }
    } else {
      for (int i = 1; i <= odd; i += 2) {
        on = on.add(choose(new BigInteger("" + odd), new BigInteger("" + i)));
      }
    }
    res = on.multiply(en);
//    System.out.println("on: " + on);
//    System.out.println("en: " + en);
    System.out.println(res.toString());
  }

  public static BigInteger[][] memo = new BigInteger[101][101];

  static BigInteger choose(BigInteger n, BigInteger k) {

    if (n.intValue() == 0 && k.intValue() > 0) {
      return BigInteger.ZERO;
    } else if (k.intValue() == 0 && n.intValue() >= 0) {
      return BigInteger.ONE;
    } else if (memo[n.intValue()][k.intValue()] != null) {
      return memo[n.intValue()][k.intValue()];
    } else {
      memo[n.intValue()][k.intValue()] = choose(n.subtract(BigInteger.ONE), k.subtract(BigInteger.ONE)).add(choose(n.subtract(BigInteger.ONE), k));
    }
    return memo[n.intValue()][k.intValue()];

  }

  // mod combination
  static long modComb(long a, long b, long mod) {
    long res = 1;
    for (long i = a; i > a - b; i--) {
      res = (res * i) % mod;
    }
    for (long i = b; i > 1; i--) {
      res = (res * inv(i, mod)) % mod;
    }
    return res;
  }

  static long inv(long a, long mod) {
    return powMod(a, mod - 2, mod);
  }

  // a ^ b (mod p)
  static long powMod(long a, long b, long p) {
    if (b == 0) {
      return 1;
    }
    if (b % 2 == 0) {
      long d = powMod(a, b / 2, p);
      return d * d % p;
    }
    return a * powMod(a, b - 1, p) % p;
  }
  static long N;

  static BigInteger factorial(int n) {
    if (n < 2) {
      return BigInteger.ONE;
    }
    BigInteger p = BigInteger.ONE, r = BigInteger.ONE;
    N = 1;
    int log2n = 31 - Integer.numberOfLeadingZeros(n);
    int h = 0, shift = 0, high = 1;
    while (h != n) {
      shift += h;
      h = n >>> log2n--;
      int len = high;
      high = (h & 1) == 1 ? h : h - 1;
      len = (high - len) >> 1;
      if (len > 0) {
        r = r.multiply((p = p.multiply(bp(len))));
      }
    }
    return r.shiftLeft(shift);
  }

  static BigInteger bp(int n) {
    int m = n >> 1;
    if (m == 0) {
      return BigInteger.valueOf(N += 2);
    } else if (n == 2) {
      return BigInteger.valueOf(N += 2).multiply(BigInteger.valueOf(N += 2));
    }
    return bp(n - m).multiply(bp(m));
  }
}