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:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find the factorial
// of a given number
long long factorial(long long num)
{
    if (num == 0 || num == 1)
        return 1;
    else
        return num * factorial(num - 1);
}

// Function to calculate the sum
// of binomial coefficients(nCr) for
// all values of r from L to R
long long sumOfnCr(int n, int R, int L)
{
    long long r;
    long long res = 0;

    for (r = L; r <= R; r++)
        res += (factorial(n)
                / (factorial(r)
                   * factorial(n - r)));

    return res;
}

// Driver Code
int main()
{
    int N = 5, L = 0, R = 3;
    cout << sumOfnCr(N, R, L);
    return 0;
}

Java

// JAVA program for the above approach
import java.io.*;

class GFG {

    // Function to find the factorial
    // of a given number
    static long factorial(long num)
    {
        if (num == 0 || num == 1)
            return 1;
        else
            return num * factorial(num - 1);
    }

    // Function to calculate the sum
    // of binomial coefficients(nCr) for
    // all values of r from L to R
    static long sumOfnCr(int n, int R, int L)
    {
        long r;
        long res = 0;

        for (r = L; r <= R; r++)
            res += (factorial(n)
                    / (factorial(r) * factorial(n - r)));

        return res;
    }

    // Driver Code
    public static void main(String[] args)
    {
        int N = 5, L = 0, R = 3;
        long ans = sumOfnCr(N, R, L);
        System.out.println(ans);
    }
}

// This code is contributed by Taranpreet

Python3

# Python code for the above approach

# Function to find the factorial
# of a given number
def factorial(num):
    if (num == 0 or num == 1):
        return 1;
    else:
        return num * factorial(num - 1);

# Function to calculate the sum
# of binomial coefficients(nCr) for
# all values of r from L to R
def sumOfnCr(n, R, L):
    res = 0;

    for r in range(L, R + 1):
        res += (factorial(n) / (factorial(r) * factorial(n - r)));

    return res;

# Driver Code
N = 5
L = 0
R = 3;
print((int)(sumOfnCr(N, R, L)))

# This code is contributed by gfgking

C#

// C# program for the above approach
using System;
class GFG {

  // Function to find the factorial
  // of a given number
  static long factorial(long num)
  {
    if (num == 0 || num == 1)
      return 1;
    else
      return num * factorial(num - 1);
  }

  // Function to calculate the sum
  // of binomial coefficients(nCr) for
  // all values of r from L to R
  static long sumOfnCr(int n, int R, int L)
  {
    long r;
    long res = 0;

    for (r = L; r <= R; r++)
      res += (factorial(n)
              / (factorial(r) * factorial(n - r)));

    return res;
  }

  // Driver Code
  public static void Main()
  {
    int N = 5, L = 0, R = 3;
    Console.Write(sumOfnCr(N, R, L));
  }
}

// This code is contributed by ukasp.

Javascript

<script>
        // JavaScript code for the above approach

        // Function to find the factorial
        // of a given number
        function factorial(num) {
            if (num == 0 || num == 1)
                return 1;
            else
                return num * factorial(num - 1);
        }

        // Function to calculate the sum
        // of binomial coefficients(nCr) for
        // all values of r from L to R
        function sumOfnCr(n, R, L) {
            let r;
            let res = 0;

            for (r = L; r <= R; r++)
                res += (factorial(n)
                    / (factorial(r)
                        * factorial(n - r)));

            return res;
        }

        // Driver Code
        let N = 5, L = 0, R = 3;
        document.write(sumOfnCr(N, R, L));

         // This code is contributed by Potta Lokesh
    </script>

Output

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.

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to calculate the sum
// of binomial coefficients(nCr) for
// all values of r from L to R
long long sumOfnCr(int n, int R, int L)
{
    long long res = 0;

    for (int r = L; r <= R; r++) {
        long long nCr = 1;
        for (int i = 1; i <= r; i++) {
            nCr *= (n - i + 1);
            nCr /= i;
        }
        res += nCr;
    }

    return res;
}

// Driver Code
int main()
{
    int N = 5, L = 0, R = 3;
    cout << sumOfnCr(N, R, L);
    return 0;
}

Java

import java.math.BigInteger;

public class BinomialCoefficientSum {

    // Function to calculate the sum of binomial coefficients (nCr)
    // for all values of r from L to R
    public static BigInteger sumOfnCr(int n, int R, int L) {
        // Initialize the result as zero
        BigInteger res = BigInteger.ZERO;

        // Iterate through all values of r from L to R
        for (int r = L; r <= R; r++) {
            // Initialize nCr as 1
            BigInteger nCr = BigInteger.ONE;

            // Calculate nCr using multiplicative formula
            for (int i = 1; i <= r; i++) {
                // Multiplicative formula for nCr: nCr = (n - i + 1) / i * nCr
                nCr = nCr.multiply(BigInteger.valueOf(n - i + 1)).divide(BigInteger.valueOf(i));
            }

            // Add the calculated nCr to the result
            res = res.add(nCr);
        }

        // Return the final result
        return res;
    }

    public static void main(String[] args) {
        // Input values
        int N = 5, L = 0, R = 3;

        // Calculate and print the sum of binomial coefficients
        System.out.println(sumOfnCr(N, R, L));
    }
}

Python3

# Function to calculate the sum
# of binomial coefficients(nCr) for
# all values of r from L to R


def sumOfnCr(n, R, L):
    res = 0
    for r in range(L, R+1):
        nCr = 1
        for i in range(1, r+1):
            nCr *= (n - i + 1)
            nCr //= i
        res += nCr
    return res


# Driver Code
N = 5
L = 0
R = 3

print(sumOfnCr(N, R, L))

C#

using System;

class Program
{
    // Function to calculate the sum of binomial coefficients(nCr) for
    // all values of r from L to R
    static long SumOfnCr(int n, int R, int L)
    {
        long res = 0;

        for (int r = L; r <= R; r++)
        {
            long nCr = 1;
            for (int i = 1; i <= r; i++)
            {
                nCr *= (n - i + 1);
                nCr /= i;
            }
            res += nCr;
        }

        return res;
    }

    static void Main(string[] args)
    {
        int N = 5, L = 0, R = 3;
        Console.WriteLine(SumOfnCr(N, R, L));
    }
}
// This code is contributed by shivamgupta310570

Javascript

// Javascript code addition

// Function to calculate the sum
// of binomial coefficients(nCr) for
// all values of r from L to R
function sumOfnCr(n, R, L) {
  let res = 0;

  for (let r = L; r <= R; r++) {
    let nCr = 1;
    for (let i = 1; i <= r; i++) {
      nCr *= (n - i + 1);
      nCr /= i;
    }
    res += nCr;
  }

  return res;
}

// Driver Code
let N = 5, L = 0, R = 3;
console.log(sumOfnCr(N, R, L));

// The code is contributed by Nidhi goel.

Output

26

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