# Sum of product of r and rth Binomial Coefficient (r * nCr)

Given a positive integer n. The task is to find the sum of the product of r and rth Binomial Coefficient. In other words find: Σ (r * nCr), where 0 <= r <= n.

Examples:

```Input : n = 2
Output : 4
0.2C0 + 1.2C1 + 2.2C2
= 0*2 + 1*2 + 2*1
= 4

Input : n = 5
Output : 80
```

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

Method 1 (Brute Force) : The idea is to iterate a loop i from 0 to n and evaluate i * nCi and add to sum variable.

Below is the implementation of this approach:

## C++

 `// CPP Program to find sum of product of r and ` `// rth Binomial Coefficient i.e summation r * nCr ` `#include ` `using` `namespace` `std; ` `#define MAX 100 ` ` `  `// Return the first n term of binomial coefficient. ` `void` `binomialCoeff(``int` `n, ``int` `C[]) ` `{ ` `    ``C = 1; ``// nC0 is 1 ` ` `  `    ``for` `(``int` `i = 1; i <= n; i++) { ` ` `  `        ``// Compute next row of pascal triangle  ` `        ``// using the previous row ` `        ``for` `(``int` `j = min(i, n); j > 0; j--) ` `            ``C[j] = C[j] + C[j - 1]; ` `    ``} ` `} ` ` `  `// Return summation of r * nCr ` `int` `summation(``int` `n) ` `{ ` `    ``int` `C[MAX]; ` `    ``memset``(C, 0, ``sizeof``(C)); ` ` `  `    ``// finding the first n term of binomial ` `    ``// coefficient ` `    ``binomialCoeff(n, C); ` ` `  `    ``// Iterate a loop to find the sum. ` `    ``int` `sum = 0; ` `    ``for` `(``int` `i = 0; i <= n; i++)  ` `        ``sum += (i * C[i]);     ` ` `  `    ``return` `sum; ` `} ` ` `  `// Driven Program ` `int` `main() ` `{ ` `    ``int` `n = 2; ` `    ``cout << summation(n) << endl; ` `    ``return` `0; ` `} `

## Java

 `// Java Program to find sum  ` `// of product of r and rth  ` `// Binomial Coefficient i.e ` `// summation r * nCr ` `class` `GFG ` `{ ` `static` `int` `MAX = ``100``; ` ` `  `// Return the first n term  ` `// of binomial coefficient. ` `static` `void` `binomialCoeff(``int` `n,  ` `                          ``int` `C[]) ` `{ ` `    ``C[``0``] = ``1``; ``// nC0 is 1 ` ` `  `    ``for` `(``int` `i = ``1``; i <= n; i++) ` `    ``{ ` ` `  `        ``// Compute next row of ` `        ``// pascal triangle using  ` `        ``// the previous row ` `        ``for` `(``int` `j = Math.min(i, n); j > ``0``; j--) ` `            ``C[j] = C[j] + C[j - ``1``]; ` `    ``} ` `} ` ` `  `// Return summation ` `// of r * nCr ` `static` `int` `summation(``int` `n) ` `{ ` `    ``int` `C[] = ``new` `int``[MAX]; ` `     `  `    ``for``(``int` `i = ``0``; i < MAX; i++) ` `    ``C[i] = ``0``; ` ` `  `    ``// finding the first n term   ` `    ``// of binomial coefficient ` `    ``binomialCoeff(n, C); ` ` `  `    ``// Iterate a loop  ` `    ``// to find the sum. ` `    ``int` `sum = ``0``; ` `    ``for` `(``int` `i = ``0``; i <= n; i++)  ` `        ``sum += (i * C[i]);  ` ` `  `    ``return` `sum; ` `} ` ` `  `// Driver Code ` `public` `static` `void` `main(String args[]) ` `{ ` `    ``int` `n = ``2``; ` `    ``System.out.println( summation(n)); ` `} ` `} ` ` `  `// This code is contributed by Arnab Kundu `

## Python 3

 `# Python 3 Program to find sum of product  ` `# of r and rth Binomial Coefficient i.e  ` `# summation r * nCr ` `MAX` `=` `100` ` `  `# Return the first n term of  ` `# binomial coefficient. ` `def` `binomialCoeff(n, C): ` ` `  `    ``C[``0``] ``=` `1` `# nC0 is 1 ` ` `  `    ``for` `i ``in` `range``(``1``, n ``+` `1``): ` ` `  `        ``# Compute next row of pascal triangle  ` `        ``# using the previous row ` `        ``for` `j ``in` `range``(``min``(i, n), ``-``1``, ``-``1``): ` `            ``C[j] ``=` `C[j] ``+` `C[j ``-` `1``] ` ` `  `# Return summation of r * nCr ` `def` `summation( n): ` ` `  `    ``C ``=` `[``0``] ``*` `MAX` ` `  `    ``# finding the first n term of  ` `    ``# binomial coefficient ` `    ``binomialCoeff(n, C) ` ` `  `    ``# Iterate a loop to find the sum. ` `    ``sum` `=` `0` `    ``for` `i ``in` `range``(n ``+` `1``):  ` `        ``sum` `+``=` `(i ``*` `C[i])  ` ` `  `    ``return` `sum` ` `  `# Driver Code ` `if` `__name__ ``=``=` `"__main__"``: ` `     `  `    ``n ``=` `2` `    ``print``(summation(n)) ` ` `  `# This code is contributed by ita_c `

## C#

 `// C# Program to find sum  ` `// of product of r and rth  ` `// Binomial Coefficient i.e ` `// summation r * nCr ` `using` `System; ` ` `  `class` `GFG ` `{ ` `static` `int` `MAX = 100; ` ` `  `// Return the first n term  ` `// of binomial coefficient. ` `static` `void` `binomialCoeff(``int` `n,  ` `                          ``int` `[]C) ` `{ ` `    ``C = 1; ``// nC0 is 1 ` ` `  `    ``for` `(``int` `i = 1; i <= n; i++) ` `    ``{ ` ` `  `        ``// Compute next row of ` `        ``// pascal triangle using  ` `        ``// the previous row ` `        ``for` `(``int` `j = Math.Min(i, n);  ` `                 ``j > 0; j--) ` `            ``C[j] = C[j] + C[j - 1]; ` `    ``} ` `} ` ` `  `// Return summation ` `// of r * nCr ` `static` `int` `summation(``int` `n) ` `{ ` `    ``int` `[]C = ``new` `int``[MAX]; ` `     `  `    ``for``(``int` `i = 0; i < MAX; i++) ` `    ``C[i] = 0; ` ` `  `    ``// finding the first n term  ` `    ``// of binomial coefficient ` `    ``binomialCoeff(n, C); ` ` `  `    ``// Iterate a loop  ` `    ``// to find the sum. ` `    ``int` `sum = 0; ` `    ``for` `(``int` `i = 0; i <= n; i++)  ` `        ``sum += (i * C[i]);  ` ` `  `    ``return` `sum; ` `} ` ` `  `// Driver Code ` `public` `static` `void` `Main() ` `{ ` `    ``int` `n = 2; ` `    ``Console.Write( summation(n)); ` `} ` `} ` ` `  `// This code is contributed  ` `// by shiv_bhakt `

## PHP

 ` 0; ``\$j``--) ` `            ``\$C``[``\$j``] = ``\$C``[``\$j``] + ``\$C``[``\$j` `- 1]; ` `    ``} ` `} ` ` `  `// Return summation of r * nCr ` `function` `summation(``\$n``) ` `{ ` `    ``global` `\$MAX``; ` `    ``\$C` `= ``array_fill``(0, ``\$MAX``, 0); ` ` `  `    ``// finding the first n term of  ` `    ``// binomial coefficient ` `    ``binomialCoeff(``\$n``, ``\$C``); ` ` `  `    ``// Iterate a loop to find the sum. ` `    ``\$sum` `= 0; ` `    ``for` `(``\$i` `= 0; ``\$i` `<= ``\$n``; ``\$i``++)  ` `        ``\$sum` `+= (``\$i` `* ``\$C``[``\$i``]);  ` ` `  `    ``return` `\$sum``; ` `} ` ` `  `// Driver Code ` `\$n` `= 2; ` `echo` `summation(``\$n``); ` ` `  `// This code is contributed by mits ` `?> `

Output:

```4
```

Method 2 (Using formula) :
Mathematically we need to find,
Σ (i * nCi), where 0 <= i <= n
= Σ (iC1 * nCi), (Since nC1 = n, we can write i as iC1)
= Σ ( (i! / (i – 1)! * 1!) * (n! / (n – i)! * i!)
On cancelling i! from numerator and denominator
= Σ (n! / (i – 1)! * (n – i)!)
= Σ n * ((n – 1)!/ (i – 1)! * (n – i)!)
(Using reverse of nCr = (n)!/ (r)! * (n – r)!)
= n * Σ n – 1Cr – 1
= n * 2n – 1 (Since Σ nCr = 2n)

Below is the implementation of this approach:

## C++

 `// CPP Program to find sum of product of r and ` `// rth Binomial Coefficient i.e summation r * nCr ` `#include ` `using` `namespace` `std; ` `#define MAX 100 ` ` `  `// Return summation of r * nCr ` `int` `summation(``int` `n) ` `{ ` `    ``return` `n << (n - 1); ` `} ` ` `  `// Driven Program ` `int` `main() ` `{ ` `    ``int` `n = 2; ` `    ``cout << summation(n) << endl; ` `    ``return` `0; ` `} `

## Java

 `// Java Program to find sum of product of ` `// r and rth Binomial Coefficient i.e  ` `// summation r * nCr ` `import` `java.io.*; ` ` `  `class` `GFG { ` ` `  `    ``static` `int` `MAX = ``100``; ` `     `  `    ``// Return summation of r * nCr ` `    ``static` `int` `summation(``int` `n) ` `    ``{ ` `        ``return` `n << (n - ``1``); ` `    ``} ` `     `  `    ``// Driven Program ` `    ``public` `static` `void` `main (String[] args) ` `    ``{ ` `        ``int` `n = ``2``; ` `        ``System.out.println( summation(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by anuj_67. `

## Python3

 `# Python3 Program to  ` `# find sum of product  ` `# of r and rth Binomial  ` `# Coefficient i.e  ` `# summation r * nCr ` ` `  `# Return summation  ` `# of r * nCr ` `def` `summation( n): ` `    ``return` `n << (n ``-` `1``); ` ` `  `# Driver Code ` `n ``=` `2``; ` `print``(summation(n)); ` ` `  `# This code is contributed  ` `# by mits `

## C#

 `// C# Program to find sum of product of ` `// r and rth Binomial Coefficient i.e  ` `// summation r * nCr ` `using` `System; ` ` `  `class` `GFG { ` ` `  `    ``//static int MAX = 100; ` `     `  `    ``// Return summation of r * nCr ` `    ``static` `int` `summation(``int` `n) ` `    ``{ ` `        ``return` `n << (n - 1); ` `    ``} ` `     `  `    ``// Driver Code ` `    ``public` `static` `void` `Main () ` `    ``{ ` `        ``int` `n = 2; ` `        ``Console.WriteLine( summation(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by anuj_67. `

## PHP

 ` `

Output:

```4
```

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