Sum of binomial coefficients (nCr) in a given range

Given three values, N, L and R, the task is to calculate the sum of binomial coefficients (ⁿC_r) for all values of r from L to R.

Examples:

Input: N = 5, L = 0, R = 3
Output: 26
Explanation: Sum of ⁵C₀ + ⁵C₁ + ⁵C₂ + ⁵C₃ = 1 + 5 + 10 + 10 = 26.

Input: N = 3, L = 3, R = 3
Output: 1

Approach(Using factorial function): Solve this problem by straightforward calculating ⁿC_r by using the formula n! / (r!(n−r)!) and calculating factorial recursively for every value of r from L to R.

Below is the implementation of the above approach:

26

Time Complexity: O(N * (R - L))
Auxiliary Space: O(N)

Approach (Without using factorial function): This approach for finding the sum of binomial coefficients (nCr) for all values of r from L to R can be implemented using two nested loops. The outer loop will iterate from L to R, and the inner loop will calculate the binomial coefficient for each value of r using the formula:

nCr = n! / (r! * (n-r)!)

where n is the given number, r is the current value of the inner loop, and ! denotes the factorial function.

The sum of all binomial coefficients can be accumulated in a variable initialized to zero before the loops start.

Steps to implement the above approach:

• Declare and initialize the variables N, L, and R  respectively.
• Call the sumOfnCr function, passing N, R, and L as arguments.
• Within the sumOfnCr function, declare a long long variable named res and initialize it to 0.
• Start a for loop with a variable r, which runs from L to R.
• Within the for loop, declare a long long variable named nCr and initialize it to 1.
• Start another for loop with a variable i, which runs from 1 to r.
• Within the inner for loop, multiply nCr by (n-i+1) and then divide it by i.
• After the inner for loop ends, add nCr to the res variable.
• After the outer for loop ends, return the res variable.
• End the sumOfnCr function.
• Print the result of the sumOfnCr function using cout.

26

Time Complexity: O(N * (R - L))
Auxiliary Space: O(N)