# Count of N-size maximum sum Arrays with elements in range [0, 2^K – 1] and Bitwise AND equal to 0

• Last Updated : 16 Jul, 2021

Given two positive integers N and K, the task is to find the number of arrays of size N such that each array element lies over the range [0, 2K – 1] with the maximum sum of array element having Bitwise AND of all array elements 0.

Examples:

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Input: N = 2 K = 2
Output: 4
Explanation:
The possible arrays with maximum sum having the Bitwise AND of all array element as 0 {0, 3}, {3, 0}, {1, 2}, {2, 1}. The count of such array is 4.

Input: N = 5 K = 6
Output: 15625

Approach: The given problem can be solved by observing the fact that as the Bitwise AND of the generated array should be 0, then for each i in the range [0, K – 1] there should be at least 1 element with an ith bit equal to 0 in its binary representation. Therefore, to maximize the sum of the array, it is optimal to have exactly 1 element with the ith bit unset.

Hence, for each of the K bits, there are NC1 ways to make it unset in 1 array element. Therefore, the resultant count of an array having the maximum sum is given by NK.

Below is the implementation of the approach :

## C++

 `// C++ program for the above approach` `#include ``using` `namespace` `std;` `// Function to find the value of X to``// the power Y``int` `power(``int` `x, unsigned ``int` `y)``{``    ``// Stores the value of X^Y``    ``int` `res = 1;` `    ``while` `(y > 0) {` `        ``// If y is odd, multiply x``        ``// with result``        ``if` `(y & 1)``            ``res = res * x;` `        ``// Update the value of y and x``        ``y = y >> 1;``        ``x = x * x;``    ``}` `    ``// Return the result``    ``return` `res;``}` `// Function to count number of arrays``// having element over the range``// [0, 2^K - 1] with Bitwise AND value``// 0 having maximum possible sum``void` `countArrays(``int` `N, ``int` `K)``{``    ``// Print the value of N^K``    ``cout << ``int``(power(N, K));``}` `// Driver Code``int` `main()``{``    ``int` `N = 5, K = 6;``    ``countArrays(N, K);` `    ``return` `0;``}`

## Java

 `// Java program for the above approach``public` `class` `GFG``{` `// Function to find the value of X to``// the power Y``static` `int` `power(``int` `x, ``int` `y)``{``  ` `    ``// Stores the value of X^Y``    ``int` `res = ``1``;` `    ``while` `(y > ``0``) {` `        ``// If y is odd, multiply x``        ``// with result``        ``if` `(y%``2``!=``0``)``            ``res = res * x;` `        ``// Update the value of y and x``        ``y = y >> ``1``;``        ``x = x * x;``    ``}` `    ``// Return the result``    ``return` `res;``}` `// Function to count number of arrays``// having element over the range``// [0, 2^K - 1] with Bitwise AND value``// 0 having maximum possible sum``static` `void` `countArrays(``int` `N, ``int` `K)``{``  ` `    ``// Print the value of N^K``   ``System.out.println((``int``)(power(N, K)));``}`  `// Driver Code``public` `static` `void` `main(String args[])``{``    ``int` `N = ``5``, K = ``6``;``    ``countArrays(N, K);``}``}` `// This code is contributed by SoumikMondal`

## Python3

 `# Python Program for the above approach` `# Function to find the value of X to``# the power Y``def` `power(x, y):``    ``# Stores the value of X^Y``    ``res ``=` `1` `    ``while` `(y > ``0``):` `        ``# If y is odd, multiply x``        ``# with result``        ``if` `(y & ``1``):``            ``res ``=` `res ``*` `x` `        ``# Update the value of y and x``        ``y ``=` `y >> ``1``        ``x ``=` `x ``*` `x` `    ``# Return the result``    ``return` `res` `# Function to count number of arrays``# having element over the range``# [0, 2^K - 1] with Bitwise AND value``# 0 having maximum possible sum``def` `countArrays(N, K):``    ``# Print the value of N^K``    ``print``(power(N, K))`  `# Driver Code` `N ``=` `5``;``K ``=` `6``;``countArrays(N, K)` `# This code is contributed by gfgking`

## C#

 `// C# program for the above approach``using` `System;` `class` `GFG{` `// Function to find the value of X to``// the power Y``static` `int` `power(``int` `x, ``int` `y)``{``    ` `    ``// Stores the value of X^Y``    ``int` `res = 1;` `    ``while` `(y > 0)``    ``{``        ` `        ``// If y is odd, multiply x``        ``// with result``        ``if` `(y % 2 != 0)``            ``res = res * x;` `        ``// Update the value of y and x``        ``y = y >> 1;``        ``x = x * x;``    ``}` `    ``// Return the result``    ``return` `res;``}` `// Function to count number of arrays``// having element over the range``// [0, 2^K - 1] with Bitwise AND value``// 0 having maximum possible sum``static` `void` `countArrays(``int` `N, ``int` `K)``{``    ` `    ``// Print the value of N^K``    ``Console.WriteLine((``int``)(power(N, K)));``}` `// Driver Code``public` `static` `void` `Main()``{``    ``int` `N = 5, K = 6;``    ` `    ``countArrays(N, K);``}``}` `// This code is contributed by subhammahato348`

## Javascript

 ``
Output:
`15625`

Time Complexity: O(log K)
Auxiliary Space: O(1)

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