Step by Step solution about How to resolve the algorithm Factorial step by step in the Mathematica / Wolfram Language programming language

The code provided is a definition of the factorial function in the Wolfram programming language. The factorial of a non-negative integer $n$ is the product of all positive integers less than or equal to $n$. For example, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.

The first definition uses recursion to compute the factorial. The function factorial takes a single argument, n, which is assumed to be a non-negative integer. The function returns $n!$ if $n > 0$, and $1$ if $n = 0$. The recursive call to factorial in the first line of the definition computes the factorial of $n-1$, and then multiplies this value by $n$ to get $n!$.

The second definition uses a Block construct to create a local variable result that is initialized to $1$. The For loop then iterates over the integers from $1$ to $n$, and in each iteration, the value of result is multiplied by the current value of i. After the loop has finished, the value of result is returned as the factorial of $n$.

The third definition uses the Times @@ Table construct to compute the factorial of $n$. The Table function generates a list of the integers from $1$ to $n$, and the Times @@ construct multiplies all of the elements in the list together. The result of this multiplication is the factorial of $n$.

All three of these definitions of the factorial function are correct, but the second definition is probably the most efficient for large values of $n$.

factorial[n_Integer] := n*factorial[n-1]
factorial[0] = 1

factorial[n_Integer] := 
  Block[{i, result = 1}, For[i = 1, i <= n, ++i, result *= i]; result]

factorial[n_Integer] := Block[{i}, Times @@ Table[i, {i, n}]]