# Sum of Binomial coefficients

Given a positive integer n, the task is to find the sum of binomail coefficient i.e

nC0 + nC1 + nC2 + ……. + nCn-1 + nCn

Examples:

```Input : n = 4
Output : 16
4C0 + 4C1 + 4C2 + 4C3 + 4C4
= 1 + 4 + 6 + 4 + 1
= 16

Input : n = 5
Output : 8
```

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

Method 1 (Brute Force):
The idea is to evaluate each binomial coefficient term i.e nCr, where 0 <= r <= n and calculate the sum of all the terms.

Below is the implementation of this approach:

## C++

 `// CPP Program to find the sum of Binomial ` `// Coefficient. ` `#include ` `using` `namespace` `std; ` ` `  `// Returns value of Binomial Coefficient Sum ` `int` `binomialCoeffSum(``int` `n) ` `{ ` `    ``int` `C[n + 1][n + 1]; ` ` `  `    ``// Calculate value of Binomial Coefficient ` `    ``// in bottom up manner ` `    ``for` `(``int` `i = 0; i <= n; i++) { ` `        ``for` `(``int` `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]; ` `        ``} ` `    ``} ` ` `  `    ``// Calculating the sum. ` `    ``int` `sum = 0; ` `    ``for` `(``int` `i = 0; i <= n; i++) ` `        ``sum += C[n][i]; ` ` `  `    ``return` `sum; ` `} ` ` `  `/* Driver program to test above function*/` `int` `main() ` `{ ` `    ``int` `n = 4; ` `    ``printf``(``"%d"``, binomialCoeffSum(n)); ` `    ``return` `0; ` `} `

## Java

 `// Java Program to find the sum  ` `// of Binomial Coefficient. ` ` `  `class` `GFG { ` `     `  `    ``// Returns value of Binomial  ` `    ``// Coefficient Sum ` `    ``static` `int` `binomialCoeffSum(``int` `n) ` `    ``{ ` `        ``int` `C[][] = ``new` `int``[n + ``1``][n + ``1``]; ` `     `  `        ``// Calculate value of Binomial  ` `        ``// Coefficient in bottom up manner ` `        ``for` `(``int` `i = ``0``; i <= n; i++) ` `        ``{ ` `            ``for` `(``int` `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]; ` `             `  `                 `  `            ``} ` `        ``} ` `     `  `        ``// Calculating the sum. ` `        ``int` `sum = ``0``; ` `        ``for` `(``int` `i = ``0``; i <= n; i++) ` `            ``sum += C[n][i]; ` `     `  `        ``return` `sum; ` `    ``} ` `     `  `    ``/* Driver program to test above function*/` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int` `n = ``4``; ` `        ``System.out.println(binomialCoeffSum(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by prerna saini. `

## Python3

 `# Python  Program to find the sum  ` `# of Binomial Coefficient. ` `  `  `import` `math     ` `  `  `# Returns value of Binomial  ` `# Coefficient Sum ` `def` `binomialCoeffSum( n): ` `     `  `        ``C ``=` `[[``0``]``*``(n``+``2``) ``for` `i ``in` `range``(``0``,n``+``2``)] ` `      `  `        ``# 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] ` `      `  `        ``# Calculating the sum. ` `        ``sum` `=` `0` `        ``for` `i ``in` `range``(``0``,n``+``1``): ` `            ``sum` `+``=` `C[n][i] ` `      `  `        ``return` `sum` `     `  `      `  `# Driver program to test above function ` `n ``=` `4` `print``(binomialCoeffSum(n)) ` ` `  `# This code is contributed by Gitanjali. `

## C#

 `// C# program to find the sum ` `// of Binomial Coefficient. ` `using` `System; ` ` `  `class` `GFG { ` ` `  `    ``// Returns value of Binomial ` `    ``// Coefficient Sum ` `    ``static` `int` `binomialCoeffSum(``int` `n) ` `    ``{ ` `        ``int``[, ] C = ``new` `int``[n + 1, n + 1]; ` ` `  `        ``// Calculate value of Binomial ` `        ``// Coefficient in bottom up manner ` `        ``for` `(``int` `i = 0; i <= n; i++)  ` `        ``{ ` `            ``for` `(``int` `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]; ` `            ``} ` `        ``} ` ` `  `        ``// Calculating the sum. ` `        ``int` `sum = 0; ` `        ``for` `(``int` `i = 0; i <= n; i++) ` `            ``sum += C[n, i]; ` ` `  `        ``return` `sum; ` `    ``} ` ` `  `    ``/* Driver program to test above function*/` `    ``public` `static` `void` `Main() ` `    ``{ ` `        ``int` `n = 4; ` `        ``Console.WriteLine(binomialCoeffSum(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by vt_m. `

## PHP

 ` `

Output:

```16
```

Method 2 (Using Formula): This can be proved in 2 ways.
First Proof: Using Principle of induction.

For basic step, n = 0
LHS = 0C0 = (0!)/(0! * 0!) = 1/1 = 1.
RHS= 20 = 1.
LHS = RHS

For induction step:
Let k be an integer such that k > 0 and for all r, 0 <= r <= k, where r belong to integers,
the formula stand true.
Therefore,
kC0 + kC1 + kC2 + ……. + kCk-1 + kCk = 2k

Now, we have to prove for n = k + 1,
k+1C0 + k+1C1 + k+1C2 + ……. + k+1Ck + k+1Ck+1 = 2k+1

LHS = k+1C0 + k+1C1 + k+1C2 + ……. + k+1Ck + k+1Ck+1
(Using nC0 = 0 and n+1Cr = nCr + nCr-1)
= 1 + kC0 + kC1 + kC1 + kC2 + …… + kCk-1 + kCk + 1
= kC0 + kC0 + kC1 + kC1 + …… + kCk-1 + kCk-1 + kCk + kCk
= 2 X ∑ nCr
= 2 X 2k
= 2k+1
= RHS

Second Proof: Using Binomial theorem expansion

Binomial expansion state,
(x + y)n = nC0 xn y0 + nC1 xn-1 y1 + nC2 xn-2 y2 + ……… + nCn-1 x1 yn-1 + nCn x0 yn

Put x = 1, y = 1
(1 + 1)n = nC0 1n 10 + nC1 xn-1 11 + nC2 1n-2 12 + ……… + nCn-1 11 1n-1 + nCn 10 1n

2n = nC0 + nC1 + nC2 + ……. + nCn-1 + nCn

Below is implementation of this approach:

## C++

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

## Java

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

## Python3

 `# Python  Program to find the sum  ` `# of Binomial Coefficient. ` `  `  `import` `math      ` `# Returns value of Binomial  ` `# Coefficient Sum ` `def` `binomialCoeffSum( n): ` `     `  `    ``return` `(``1` `<< n); ` ` `  `# Driver program to test ` `# above function ` `n ``=` `4` `print``(binomialCoeffSum(n)) ` ` `  `# This code is contributed ` `# by Gitanjali. `

## C#

 `// C# Program to find sum of  ` `// Binomial Coefficient. ` `using` `System; ` ` `  `class` `GFG { ` ` `  `    ``// Returns value of Binomial Coefficient Sum ` `    ``// which is 2 raised to power n. ` `    ``static` `int` `binomialCoeffSum(``int` `n) ` `    ``{ ` `        ``return` `(1 << n); ` `    ``} ` ` `  `    ``/* Drier program to test above function*/` `    ``static` `public` `void` `Main() ` `    ``{ ` `        ``int` `n = 4; ` `        ``Console.WriteLine(binomialCoeffSum(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by vt_m. `

## PHP

 ` `

Output:

```16
```

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