Sum of squares of binomial coefficients

Last Updated : 28 Mar, 2023

Given a positive integer n. The task is to find the sum of square of Binomial Coefficient i.e 
nC02 + nC12 + nC22 + nC32 + ......... + nCn-22 + nCn-12 + nCn2 
Examples: 

Input : n = 4
Output : 70

Input : n = 5
Output : 252

Method 1: (Brute Force) 
The idea is to generate all the terms of binomial coefficient and find the sum of square of each binomial coefficient.
Below is the implementation of this approach:

C++ 
// CPP Program to find the sum of square of 
// binomial coefficient.
#include<bits/stdc++.h>
using namespace std;

// Return the sum of square of binomial coefficient
int sumofsquare(int n)
{
    int C[n+1][n+1];
    int i, j;

    // Calculate value of Binomial Coefficient
    // in bottom up manner
    for (i = 0; i <= n; i++)
    {
        for (j = 0; j <= min(i, n); j++)
        {
            // Base Cases
            if (j == 0 || j == i)
                C[i][j] = 1;

            // Calculate value using previously
            // stored values
            else
                C[i][j] = C[i-1][j-1] + C[i-1][j];
        }
    }    
    
    // Finding the sum of square of binomial 
    // coefficient.
    int sum = 0;
    for (int i = 0; i <= n; i++)    
        sum += (C[n][i] * C[n][i]);    
    
    return sum;
}

// Driven Program
int main()
{
    int n = 4;    
    cout << sumofsquare(n) << endl;
    return 0;
}
Java 
// Java Program to find the sum of 
// square of binomial coefficient.
import static java.lang.Math.*;

class GFG{
        
    // Return the sum of square of 
    // binomial coefficient
    static int sumofsquare(int n)
    {
        int[][] C = new int [n+1][n+1] ;
        int i, j;
    
        // Calculate value of Binomial 
        // Coefficient in bottom up manner
        for (i = 0; i <= n; i++)
        {
            for (j = 0; j <= min(i, n); j++)
            {
                // Base Cases
                if (j == 0 || j == i)
                    C[i][j] = 1;
    
                // Calculate value using 
                //previously stored values
                else
                    C[i][j] = C[i-1][j-1] 
                             + C[i-1][j];
            }
        } 
        
        // Finding the sum of square of 
        // binomial coefficient.
        int sum = 0;
        for (i = 0; i <= n; i++) 
            sum += (C[n][i] * C[n][i]); 
        
        return sum;
    }
    
    // Driver function
    public static void main(String[] args)
    {
        int n = 4; 
        
        System.out.println(sumofsquare(n));
    } 
}

// This code is contributed by 
// Smitha Dinesh Semwal
Python3 
# Python Program to find 
# the sum of square of
# binomial coefficient.

# Return the sum of 
# square of binomial
# coefficient
def sumofsquare(n) :
    
    C = [[0 for i in range(n + 1)] 
            for j in range(n + 1)]
            
    # Calculate value of 
    # Binomial Coefficient 
    # in bottom up manner
    for i in range(0, n + 1) :
    
        for j in range(0, min(i, n) + 1) :
                    
            # Base Cases
            if (j == 0 or j == i) :
                C[i][j] = 1

            # Calculate value 
            # using previously
            # stored values
            else :
                C[i][j] = (C[i - 1][j - 1] +
                           C[i - 1][j])

    
    # Finding the sum of 
    # square of binomial 
    # coefficient.
    sum = 0
    for i in range(0, n + 1) :
        sum = sum + (C[n][i] * 
                     C[n][i]) 
    
    return sum


# Driver Code
n = 4
print (sumofsquare(n), end="\n")
    
# This code is contributed by 
# Manish Shaw(manishshaw1)
C# 
// C# Program to find the sum of 
// square of binomial coefficient.
using System;

class GFG {
        
    // Return the sum of square of 
    // binomial coefficient
    static int sumofsquare(int n)
    {
        int[,] C = new int [n+1,n+1] ;
        int i, j;
    
        // Calculate value of Binomial 
        // Coefficient in bottom up manner
        for (i = 0; i <= n; i++)
        {
            for (j = 0; j <= Math.Min(i, n); j++)
            {
                // Base Cases
                if (j == 0 || j == i)
                    C[i,j] = 1;
    
                // Calculate value using 
                //previously stored values
                else
                    C[i,j] = C[i-1,j-1] 
                            + C[i-1,j];
            }
        } 
        
        // Finding the sum of square of 
        // binomial coefficient.
        int sum = 0;
        for (i = 0; i <= n; i++) 
            sum += (C[n,i] * C[n,i]); 
        
        return sum;
    }
    
    // Driver function
    public static void Main()
    {
        int n = 4; 
        
        Console.WriteLine(sumofsquare(n));
    } 
}

// This code is contributed by vt_m.
PHP 
<?php
// PHP Program to find the sum of
// square of binomial coefficient.

// Return the sum of square of binomial
// coefficient
function sumofsquare($n)
{
    $i; $j;

    // Calculate value of Binomial
    // Coefficient in bottom up manner
    for ($i = 0; $i <= $n; $i++)
    {
        for ($j = 0; $j <= min($i, $n); $j++)
        {
            
            // Base Cases
            if ($j == 0 || $j == $i)
                $C[$i][$j] = 1;

            // Calculate value using previously
            // stored values
            else
                $C[$i][$j] = $C[$i-1][$j-1]
                                + $C[$i-1][$j];
        }
    } 
    
    // Finding the sum of square of binomial 
    // coefficient.
    $sum = 0;
    for ($i = 0; $i <= $n; $i++) 
        $sum += ($C[$n][$i] * $C[$n][$i]); 
    
    return $sum;
}

// Driven Program
    $n = 4; 
    echo sumofsquare($n), "\n";
    
// This code is contributed by ajit
?>
JavaScript 
<script>

// JavaScript Program to find the sum of 
// square of binomial coefficient.

    // Return the sum of square of 
    // binomial coefficient
    function sumofsquare(n)
    {
        let  C = new Array(n+1); 
        
        // Loop to create 2D array using 1D array
        for (let i = 0; i < C.length; i++) {
            C[i] = new Array(2);
        }
        
        let  i, j;
      
        // Calculate value of Binomial 
        // Coefficient in bottom up manner
        for (i = 0; i <= n; i++)
        {
            for (j = 0; j <= Math.min(i, n); j++)
            {
            
                // Base Cases
                if (j == 0 || j == i)
                    C[i][j] = 1;
      
                // Calculate value using 
                //previously stored values
                else
                    C[i][j] = C[i-1][j-1] 
                             + C[i-1][j];
            }
        } 
          
        // Finding the sum of square of 
        // binomial coefficient.
        let sum = 0;
        for (i = 0; i <= n; i++) 
            sum += (C[n][i] * C[n][i]); 
          
        return sum;
    }

// Driver code     
        let  n = 4; 
        document.write(sumofsquare(n));
       
       // This code is contributed by code_hunt.
</script>

Output:  

70

Time Complexity: O(n2)
Space Complexity: O(n2)


Method 2: (Using Formula) 
^nC^2_0 + ^nC^2_1 + ^nC^2_2 + .... + ^nC^2_n-1 + ^nC^2_n        
= ^2nC_n        
= \frac{(2n)!}{(n!)^2}        
Proof, 
 

We know,
(1 + x)n = nC0 + nC1 x + nC2 x2 + ......... + nCn-1 xn-1 + nCn-1 xn
Also,
(x + 1)n = nC0 xn + nC1 xn-1 + nC2 xn-2 + ......... + nCn-1 x + nCn

Multiplying above two equations,
(1 + x)2n = [nC0 + nC1 x + nC2 x2 + ......... + nCn-1 xn-1 + nCn-1 xn] X 
            [nC0 xn + nC1 xn-1 + nC2 xn-2 + ......... + nCn-1 x + nCn]

Equating coefficients of xn on both sides, we get
2nCn = nC02 + nC12 + nC22 + nC32 + ......... + nCn-22 + nCn-12 + nCn2

Hence, sum of the squares of coefficients = 2nCn = (2n)!/(n!)2.


Also, (2n)!/(n!)2 = (2n * (2n - 1) * (2n - 2) * ..... * (n+1))/(n * (n - 1) * (n - 2) *..... * 1).
Below is the implementation of this approach: 
 

C++ 
// CPP Program to find the sum of square of 
// binomial coefficient.
#include<bits/stdc++.h>
using namespace std;

// function to return product of number 
// from start to end.
int factorial(int start, int end)
{
    int res = 1;
    for (int i = start; i <= end; i++)
        res *= i;
        
    return res;
}

// Return the sum of square of binomial
// coefficient
int sumofsquare(int n)
{
    return factorial(n+1, 2*n)/factorial(1, n);
}

// Driven Program
int main()
{
    int n = 4;    
    cout << sumofsquare(n) << endl;
    return 0;
}
Java 
// Java Program to find the sum of square of 
// binomial coefficient.
class GFG{
    
    // function to return product of number 
    // from start to end.
    static int factorial(int start, int end)
    {
        int res = 1;
        for (int i = start; i <= end; i++)
            res *= i;
            
        return res;
    }
    
    // Return the sum of square of binomial
    // coefficient
    static int sumofsquare(int n)
    {
        return factorial(n+1, 2*n)/factorial(1, n);
    }
    
    // Driven Program
    public static void main(String[] args)
    {
        int n = 4; 
        System.out.println(sumofsquare(n));
    } 
} 

// This code is contributed by
// Smitha DInesh Semwal
Python 
# Python 3 Program to find the sum of 
# square of binomial coefficient.

# function to return product of number 
# from start to end.
def factorial(start, end):

    res = 1
    
    for i in range(start, end + 1):
        res *= i
        
    return res

# Return the sum of square of binomial
# coefficient
def sumofsquare(n):

    return int(factorial(n + 1, 2 * n)
                     /factorial(1, n))

# Driven Program

n = 4
print(sumofsquare(n))


# This code is contributed by
# Smitha Dinesh Semwal
C# 
// C# Program to find the sum of square of 
// binomial coefficient.
using System;

class GFG {
    
    // function to return product of number 
    // from start to end.
    static int factorial(int start, int end)
    {
        int res = 1;
        
        for (int i = start; i <= end; i++)
            res *= i;
            
        return res;
    }
    
    // Return the sum of square of binomial
    // coefficient
    static int sumofsquare(int n)
    {
        return factorial(n+1, 2*n)/factorial(1, n);
    }
    
    // Driven Program
    public static void Main()
    {
        int n = 4; 
        
        Console.WriteLine(sumofsquare(n));
    } 
} 

// This code is contributed by vt_m.
PHP 
<?php
// PHP Program to find the sum 
// of square of binomial coefficient.

// function to return 
// product of number 
// from start to end.
function factorial($start, $end)
{
    $res = 1;
    for ($i = $start; 
         $i <= $end; $i++)
        $res *= $i;
        
    return $res;
}

// Return the sum of 
// square of binomial
// coefficient
function sumofsquare($n)
{
    return factorial($n + 1, 
                     2 * $n) / 
           factorial(1, $n);
}

// Driver Code
$n = 4; 
echo sumofsquare($n), "\n";
    
// This code is contributed by ajit
?>
JavaScript 
<script>

    // Javascript Program to find
    // the sum of square of
    // binomial coefficient.
    
    // function to return product of number
    // from start to end.
    function factorial(start, end)
    {
        let res = 1;
         
        for (let i = start; i <= end; i++)
            res *= i;
             
        return res;
    }
     
    // Return the sum of square of binomial
    // coefficient
    function sumofsquare(n)
    {
        return parseInt
        (factorial(n+1, 2*n)/factorial(1, n), 10);
    }
    
    let n = 4;
         
      document.write(sumofsquare(n));
    
</script>

Output:  

70


Time Complexity: O(n)
Auxiliary Space: O(1)

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