Step by Step solution about How to resolve the algorithm Evaluate binomial coefficients step by step in the Ruby programming language

This Ruby code defines a method called choose for the Integer class, which calculates the binomial coefficient of two numbers. The binomial coefficient represents the number of ways to choose k elements from a set of n elements, and it is often denoted as n choose k or nCk.

Here's a step-by-step explanation of the code:

1. Integer#choose Method: The choose method is defined as an instance method of the Integer class, which means it can be called on any integer object. It takes one argument, k, which represents the number of elements to choose.

2. Calculating n choose k: Within the choose method, the binomial coefficient is calculated as n!/(n-k)!/k!. This formula follows the definition of the binomial coefficient, which is n! divided by the product of (n-k)! and k!.

3. **Computing n!: The inject method is used to compute the numerator n!. It takes an initial value of 1 and iterates over the range (self-k+1 .. self) to multiply all the integers in that range. This gives the value of n!.

4. **Computing k!: The inject method is also used to compute the denominator k!. It iterates over the range (2 .. k) and multiplies all the integers in that range. This gives the value of k!.

5. Division: Finally, the calculated n! and k! are divided to obtain the binomial coefficient n choose k.

6. c Method: Additionally, a separate method called c is defined. This method also calculates the binomial coefficient using a slightly different formula. It iterates over a range from 0 to r-1 and multiplies (n - i) by (m * (n - i)) / (i + 1). Here, m accumulates the result, n is the total number of elements, r is the number of elements to choose, and i is a loop variable.

7. Combination Count: The code also uses the combination method of the Range class to generate all possible combinations of 30 elements from a set of 1 to 60. The size method is then called on the resulting array of combinations to count the total number of combinations.

8. Usage: The choose method and the combination method are demonstrated with a few examples. The binomial coefficient 5 choose 3 and 60 choose 30 are calculated and printed. The number of ways to choose 30 out of the first 60 natural numbers is also calculated and printed using the combination method.

class Integer
  # binomial coefficient: n C k
  def choose(k)
    # n!/(n-k)!
    pTop = (self-k+1 .. self).inject(1, &:*) 
    # k!
    pBottom = (2 .. k).inject(1, &:*)
    pTop / pBottom
  end
end

p 5.choose(3)
p 60.choose(30)


def c n, r
  (0...r).inject(1) do |m,i| (m * (n - i)) / (i + 1) end
end


(1..60).to_a.combination(30).size  #=> 118264581564861424