Program to calculate value of nCr

Given two numbers n and r, The task is to find the value of ⁿC_r . Combinations represent the number of ways to choose r elements from a set of n distinct elements, without regard to the order in which they are selected. The formula for calculating combinations is :

Note: If r is greater than n, return 0.

Examples

Input: n = 5, r = 2
Output: 10 
Explanation: The value of ⁵C₂ is calculated as 5! / ((5−2)! * 2!​)= 10.

Input: n = 2, r = 4
Output: 0
Explanation: Since r is greater than n, thus ²C₄ = 0

Input: n = 5, r = 0
Output: 1
Explanation: The value of ⁵C₀ is calculated as 5!/(5−0)!*0! = 5!/5!*0! = 1.

[Naive Approach] Using Recursion - O(2^n) Time and O(n) Space

The idea is to use a recursive function to calculate the value of ⁿC_r. The base cases are:

• if r is greater than n, return 0 (there are no combinations possible)
• if r is 0 or r is n, return 1 (there is only 1 combination possible in these cases)

For other values of n and r, the function calculates the value of ⁿC_r by adding the number of combinations possible by including the current element and the number of combinations possible by not including the current element. 

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[Better Approach - 1] Using Factorial - O(n) Time and O(1) Space

This approach calculates the ⁿC_r using the factorial formula. It first computes the factorial of a given number by multiplying all integers from 1 to that number.
To find ⁿC_r, it calculates the factorial of n, r, and (n - r) separately, then applies the formula n! / (r!(n-r)!) to determine the result. Since factorial values grow rapidly, this method is inefficient for large values due to integer overflow and excessive computations.

Note: This approach may produce incorrect results due to integer overflow when handling large values of n and r.

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[Better Approach - 2] Avoiding Factorial Computations - O(n) Time and O(1) Space

The formula for ⁿC_r is n! / (r!(n-r)!).
Instead of computing full factorials, we avoid redundant calculations by recognizing that r! and (n-r)! share common terms that cancel out.
To optimize, we compute the product of numbers from r+1 to n and divide it by the product of numbers from 1 to (n-r).
Here, r is chosen as the maximum of r and (n-r) to reduce the number of multiplications.
This approach avoids large factorial values, reducing computational overhead and preventing integer overflow.

Note: This approach may produce incorrect results due to integer overflow when handling large values of n and r.

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[Expected Approach] By using Binomial Coefficient formula - O(r) Time and O(1) Space

A binomial coefficient C(n, k) can be defined as the coefficient of X^k in the expansion of (1 + X)ⁿ.
A binomial coefficient C(n, k) also gives the number of ways, disregarding order, that k objects can be chosen from among n objects; more formally, the number of k-element subsets (or k-combinations) of an n-element set.
Iterative way of calculating ⁿC_r   using binomial coefficient formula.

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[Alternate Approach] Using Logarithmic Formula - O(r) Time and O(1)

Logarithmic formula for ⁿC_r is an alternative to the factorial formula that avoids computing factorials directly and it's more efficient for large values of n and r. It uses the identity log(n!) = log(1) + log(2) + ... + log(n) to express the numerator and denominator of the ⁿC_r in terms of sums of logarithms which allows to calculate the ⁿC_r using the Logarithmic operations. This approach is faster and very efficient.
The logarithmic formula for ⁿC_r is: ⁿC_r = exp( log(n!) - log(r!) - log((n-r)!))

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