Given a range [low, high], both inclusive, and an integer K, the task is to select K numbers from the range(a number can be chosen multiple times) such that the sum of those K numbers is even. Print the number of all such permutations.

Examples:

Input:_ low = 4, high = 5, k = 3_

Output:_ 4_

Explanation:

There are 4 valid permutation. They are {4, 4, 4}, {4, 5, 5}, {5, 4, 5} and {5, 5, 4} which sum up to an even number.

Input:_ low = 1, high = 10, k = 2_

Output:_ 50_

Explanation:

There are 50 valid permutations. They are {1, 1}, {1, 3}, … {1, 9} {2, 2}, {2, 4}, …, {2, 10}, …, {10, 2}, {10, 4}, … {10, 10}.

These 50 permutations, each sum up to an even number.

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Naive Approach: The idea is to find all subset of size K such that the sum of the subset is even and also calculate permutation for each required subset.

Time Complexity:_ O(K * (2K))_

Auxiliary Space:_ O(K)_

Efficient Approach: The idea is to use the fact that the sum of two even and odd numbers is always even. Follow the steps below to solve the problem:

  1. Find the total count of even and odd numbers in the given range [low, high].
  2. Initialize variable even_sum = 1 and odd_sum = 0 to store way to get even sum and odd sum respectively.
  3. Iterate a loop K times and store the previous even sum as prev_even = even_sum and the previouse odd sum as prev_odd = odd_sum where even_sum = (prev_eveneven_count) + (prev_oddodd_count) and odd_sum = (prev_evenodd_count) + (prev_oddeven_count).
  4. Print the even_sum at the end as there is a count for odd sum because the previous odd_sum will contribute to the next even_sum.

Below is the implementation of the above approach:

  • Java

// Java program for the above approach

**import** java.util.*;

**class** GFG {

// Function to return the number

// of all permutations such that

// sum of K numbers in range is even

**public** **static** **void**

countEvenSum(``**int** low, **int** high,

**int** k)

{

// Find total count of even and

// odd number in given range

**int** even_count = high / 2 - (low - 1``) / 2``;

**int** odd_count = (high + 1``) / 2 - low / 2``;

**long** even_sum = 1``;

**long** odd_sum = 0``;

// Iterate loop k times and update

// even_sum & odd_sum using

// previous values

**for** (``**int** i = 0``; i < k; i++) {

// Update the prev_even and

// odd_sum

**long** prev_even = even_sum;

**long** prev_odd = odd_sum;

// Even sum

even_sum = (prev_even * even_count)

+ (prev_odd * odd_count);

// Odd sum

odd_sum = (prev_even * odd_count)

+ (prev_odd * even_count);

}

// Return even_sum

System.out.println(even_sum);

}

// Driver Code

**public** **static** **void** main(String[] args)

{

// Given ranges

**int** low = 4``;

**int** high = 5``;

// Length of permutation

**int** K = 3``;

// Function call

countEvenSum(low, high, K);

}

}

Output:

4

Time Complexity:_ O(K)_

Auxiliary Space:_ O(1)_

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Count of permutations such that sum of K numbers from given range is even
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