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Binomial coefficient using C++

 We are aware of the Binomial coefficient, which is a mathematical term used to represent the number of ways to select a certain number of items/objects from a larger set, without regard to their order. It's denoted by a symbol C(n, k)(n choose k or n over k).

It gets computed by following formula: C(n, k) = n! / (k! * (n - k)!) 

The C++ implementation using the above formula is: 

int factorial(int n) {
    if (n <= 1)
        return 1;
    else
        return n * factorial(n - 1);
}

int binomialCoefficient(int n, int r)
{
    if (r > n)
        return 0;

    int numerator = factorial(n);
    int denominator = factorial(r) * factorial(n - r);
    int result = numerator / denominator;

    return result;
}
int main()
{
    std::cout << "Value of C(" << 5 << ", " << 2 << ") is "
         << binomialCoefficient(5, 2) << "\n";
}


So, the number of ways to choose two items/objects from a set of 5 objects is equal to 

            5!/2! * 3! = 5*4*3*2*1 / (2*1)(3*2*1) = 5 * 2 = 10


When dealing with combinations, the order doesn't matter, however, each item or object can be selected only once. In case we need to select items or objects multiple times then we need to use a multiset coefficient.

Anyway, this text is only on the Binomial coefficient. The very standard implementation of finding a Binomial coefficient is recursion using the following formula:

            C(n, k) = C(n-1, k-1) + C(n-1, k)

            C(n, 0) = C(n, n) = 1 // This is base case

The standard implementation of C++ code is like below:

#include <iostream>
#include <vector>

int binomCoefficient(int n, int k)
{
    if(k > n) return 0;
    if((k == 0) || (k == n)) return 1;

    return binomCoefficient(n - 1, k - 1) + binomCoefficient(n - 1, k);
}

int main()
{
    std::cout << "Value of C(" << 5 << ", " << 2 << ") is "
         << binomCoefficient(5, 2) << "\n";
}

So, the function int binomCoefficient(int n, int k) is getting called recursively and
we can see there are many overlapping subproblems that actually got solved but getting
calculated again over recursive calls.

The Time & space complexity is O(n*max(k,n-k)).

Can we improve this not by calling the same subproblem? And the answer is certainly we do. We can create a 2D array like any other DP problem and store already computed subproblems' results into it in a bottom-up manner. Let's see the code:

#include <iostream>
#include <vector>

int binomCoefficientOverlapping(int n, int k)
{
    std::vector<std::vector<int>> C(n + 1, std::vector<int>(k + 1));
   
    for(int i = 0; i <= n; ++i)
    {
        for(int j = 0; j <= std::min(i, k); ++j)
        {
            if(j == 0 or j == i) // This is for base cases
                C[i][j] = 1;
            else
                // Computing from already computed subproblems
                C[i][j] = C[i - 1][j - 1] + C[i - 1][j];
        }
    }

    return C[n][k];
}

int main()
{
    std::cout << "Value of C(" << 5 << ", " << 2 << ") is "
         << binomCoefficientOverlapping(5, 2) << "\n";
   
}

The Time & space complexity is O(n*k). This can be further optimized for space. However, I am not going in that direction, rather was curious to know if is there any standard C++ function that can calculate Binomial Coefficient for me. Then I found a function in the standard template library and it's named std::beta (source)[C++17].
Binomial coefficients can be expressed in terms of the beta function:

C(n, k) = 1/((n+1) * std::beta(n-k+1,k+1));

Now, in C++ with the help of the std::beta function calculating the
binomial coefficient became just a line of code. Let's have a look at the code:

#include <iostream>
#include <cmath>

double binom(int n, int k) {
    return 1/((n+1)*std::beta(n-k+1,k+1));
}

int main()
{
std::cout << "Value of C(" << 5 << ", " << 2 << ") is "
<< binom(5, 2);
}

The beta function implementation depends on the underlying gamma functions implementation.
The beta function implementation is like the below:

beta(x, y) = Γ(x) * Γ(y) / Γ(x + y), Γ(z)represents the gamma function.

Reference: Beta_Function
The Demo.

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