# Find sum of even index binomial coefficients

Last Updated : 24 Sep, 2022

Given a positive integer n. The task is to find the sum of even indexed binomial coefficient. That is,
nC0 + nC2 + nC4 + nC6 + nC8 + ………..
Examples :

Input : n = 4
Output : 8
Explanation:
4C0 + 4C2 + 4C4
= 1 + 6 + 1
= 8

Input : n = 6
Output : 32

Method 1: (Brute Force)
The idea is to find all the binomial coefficients and find only the sum of even indexed values.

## CPP

 `// CPP Program to find sum ``// of even index term``#include ``using` `namespace` `std;` `// Return the sum of ``// even index term``int` `evenSum(``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 sum of even index term.``    ``int` `sum = 0;``    ``for` `(``int` `i = 0; i <= n; i += 2)``        ``sum += C[n][i];` `    ``return` `sum;``}` `// Driver Program``int` `main()``{``    ``int` `n = 4;``    ``cout << evenSum(n) << endl;``    ``return` `0;``}`

## Java

 `// Java Program to find sum ``// of even index term``import` `java.io.*;``import` `java.math.*;` `class` `GFG {``    ` `    ``// Return the sum of ``    ``// even index term``    ``static` `int` `evenSum(``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``;``     ` `                ``// else Calculate value using ``                ``// previously stored values``                ``else``                    ``C[i][j] = C[i - ``1``][j - ``1``] ``                                ``+ C[i - ``1``][j];``            ``}``        ``}    ``     ` `        ``// finding sum of even index term.``        ``int` `sum = ``0``;``        ``for` `(i = ``0``; i <= n; i += ``2``)``            ``sum += C[n][i];``     ` `        ``return` `sum;``    ``}``     ` `    ``// Driver Program``    ``public` `static` `void` `main(String args[])``    ``{``        ``int` `n = ``4``;``        ``System.out.println(evenSum(n));``    ``}``}` `/*This code is contributed by Nikita Tiwari.*/`

## Python

 `# Python Program to find sum of even index term``import` `math ` `# Return the sum of even index term``def` `evenSum(n) :``    ``# Creates a list containing n+1 lists,``    ``# each of n+1 items, all set to 0``    ``C ``=` `[[``0` `for` `x ``in` `range``(n ``+` `1``)] ``for` `y ``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 sum of even index term``    ``sum` `=` `0``;``    ``for` `i ``in` `range``(``0``, n ``+` `1``):``        ``if` `n ``%` `2` `=``=` `0``:``            ``sum` `=` `sum` `+` `C[n][i]``            ` `    ``return` `sum``    ` `# Driver method``n ``=` `4``print` `evenSum(n)`  `# This code is contributed by 'Gitanjali'.`

## C#

 `// C# Program to find sum ``// of even index term``using` `System;` `class` `GFG {``    ` `    ``// Return the sum of ``    ``// even index term``    ``static` `int` `evenSum(``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;``    ` `                ``// else Calculate value using ``                ``// previously stored values``                ``else``                    ``C[i,j] = C[i - 1,j - 1] ``                            ``+ C[i - 1,j];``            ``}``        ``} ``    ` `        ``// finding sum of even index term.``        ``int` `sum = 0;``        ``for` `(i = 0; i <= n; i += 2)``            ``sum += C[n,i];``    ` `        ``return` `sum;``    ``}``    ` `    ``// Driver Program``    ``public` `static` `void` `Main()``    ``{``        ``int` `n = 4;``        ``Console.WriteLine(evenSum(n));``    ``}``}` `/*This code is contributed by vt_m.*/`

## PHP

 ``

## Javascript

 `    `

Output
```8
```

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

Method 2: (Using Formula)
Sum of even indexed binomial coefficient :

Proof :

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

Now put x = -x, we get
(1 - x)n = nC0 - nC1 x + nC2 x2 + ..... + (-1)n nCn xn

Now, adding both the above equation, we get,
(1 + x)n + (1 - x)n = 2 * [nC0 + nC2 x2 + nC4 x4 + .......]

Put x = 1
(1 + 1)n + (1 - 1)n = 2 * [nC0 + nC2 + nC4 + .......]
2n/2 = nC0 + nC2 + nC4 + .......
2n-1 = nC0 + nC2 + nC4 + .......```

Below is the implementation of this approach :

## C++

 `// CPP Program to find sum even indexed Binomial``// Coefficient.``#include ``using` `namespace` `std;` `// Returns value of even indexed Binomial Coefficient``// Sum which is 2 raised to power n-1.``int` `evenbinomialCoeffSum(``int` `n)``{``    ``return` `(1 << (n - 1));``}` `/* Driver program to test above function*/``int` `main()``{``    ``int` `n = 4;``    ``printf``(``"%d"``, evenbinomialCoeffSum(n));``    ``return` `0;``}`

## Java

 `// Java Program to find sum even indexed ``// Binomial Coefficient.``import` `java.io.*;` `class` `GFG {``// Returns value of even indexed Binomial Coefficient``// Sum which is 2 raised to power n-1.``static` `int` `evenbinomialCoeffSum(``int` `n)``{``    ``return` `(``1` `<< (n - ``1``));``}` `// Driver Code``public` `static` `void` `main(String[] args)``{``int` `n = ``4``;``    ``System.out.println(evenbinomialCoeffSum(n));``}``    ``}` `// This code is contributed by 'Gitanjali'.`

## Python

 `# Python program to find sum even indexed ``# Binomial Coefficient``import` `math ` `# Returns value of even indexed Binomial Coefficient``# Sum which is 2 raised to power n-1.``def` `evenbinomialCoeffSum( n):` `    ``return` `(``1` `<< (n ``-` `1``))` `# Driver method``if` `__name__ ``=``=` `'__main__'``:``    ``n ``=` `4``    ``print` `evenbinomialCoeffSum(n)` `# This code is contributed by 'Gitanjali'.`

## C#

 `// C# Program to find sum even indexed ``// Binomial Coefficient.``using` `System;` `class` `GFG ``{``    ``// Returns value of even indexed ``    ``// Binomial Coefficient Sum which ``    ``// is 2 raised to power n-1.``    ``static` `int` `evenbinomialCoeffSum(``int` `n)``    ``{``        ``return` `(1 << (n - 1));``    ``}``    ` `    ``// Driver Code``    ``public` `static` `void` `Main()``    ``{``        ``int` `n = 4;``        ``Console.WriteLine(evenbinomialCoeffSum(n));``    ``}``}` `// This code is contributed by 'Vt_m'.`

## PHP

 ``

## Javascript

 ``

Output
`8`

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

Sum of odd index binomial coefficient
Using the above result we can easily prove that the sum of odd index binomial coefficient is also 2n-1.

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