# Sum of squares of binomial coefficients

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


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

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  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

 

Output:

70


Method 2: (Using Formula) = = 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  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

 

Output:

70


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