Given a positive integer n, the task is to find the sum of binomial 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 : 32
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:
// CPP Program to find the sum of Binomial // Coefficient. #include <bits/stdc++.h> 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 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. |
# 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# 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 // PHP Program to find the // sum of Binomial Coefficient. // Returns value of Binomial // Coefficient Sum function binomialCoeffSum( $n )
{ $C [ $n + 1][ $n + 1] = array (0);
// 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 ];
}
}
// Calculating the sum.
$sum = 0;
for ( $i = 0; $i <= $n ; $i ++)
$sum += $C [ $n ][ $i ];
return $sum ;
} // Driver Code $n = 4;
echo binomialCoeffSum( $n );
// This code is contributed by ajit ?> |
<script> // JavaScript Program to find the sum // of Binomial Coefficient. // Returns value of Binomial
// Coefficient Sum
function binomialCoeffSum(n)
{
let C = new Array(n + 1);
// Loop to create 2D array using 1D array
for ( var i = 0; i < C.length; i++) {
C[i] = new Array(2);
}
// Calculate value of Binomial
// Coefficient in bottom up manner
for (let i = 0; i <= n; i++)
{
for (let 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.
let sum = 0;
for (let i = 0; i <= n; i++)
sum += C[n][i];
return sum;
}
// Driver code let n = 4;
document.write(binomialCoeffSum(n));
</script> |
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:
// CPP Program to find sum of Binomial // Coefficient. #include <bits/stdc++.h> using namespace std;
// Returns value of Binomial Coefficient Sum // which is 2 raised to power n. int binomialCoeffSum( int n)
{ return (1 << n);
} /* Driver program to test above function*/ int main()
{ int n = 4;
printf ( "%d" , binomialCoeffSum(n));
return 0;
} |
// 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. |
# 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# 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);
}
/* Driver program to test above function*/
static public void Main()
{
int n = 4;
Console.WriteLine(binomialCoeffSum(n));
}
} // This code is contributed by vt_m. |
<?php // PHP Program to find sum // of Binomial Coefficient. // Returns value of Binomial // Coefficient Sum which is // 2 raised to power n. function binomialCoeffSum( $n )
{ return (1 << $n );
} // Driver Code $n = 4;
echo binomialCoeffSum( $n );
// This code is contributed // by akt_mit ?> |
<script> // Javascript Program to find sum of Binomial Coefficient.
// Returns value of Binomial Coefficient Sum
// which is 2 raised to power n.
function binomialCoeffSum(n)
{
return (1 << n);
}
let n = 4;
document.write(binomialCoeffSum(n));
</script> |
Output:
16