Question 1

What is a factorial?

Accepted Answer

A factorial, written n!, is the product of all positive integers up to n. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120. By convention, 0! = 1. Factorials count the number of ways to arrange n distinct objects in a sequence (there are n! different permutations).

Question 2

Why does 0! equal 1?

Accepted Answer

The convention 0! = 1 is mathematically necessary for consistency. There is exactly one way to arrange zero objects: do nothing. It also ensures formulas like nCr = n! / (r!(n-r)!) work correctly when r = 0 or r = n. Additionally, the recurrence n! = n x (n-1)! requires 0! = 1 for the case n = 1.

Question 3

What is the difference between permutations and combinations?

Accepted Answer

Permutations count arrangements where order matters: how many ways can you arrange r items from n? (nPr = n!/(n-r)!). Combinations count selections where order does not matter: how many ways can you choose r items from n? (nCr = n!/(r!(n-r)!)). For example, from {A,B,C}: permutations of 2 are AB, BA, AC, CA, BC, CB (6), but combinations of 2 are AB, AC, BC (3).

Question 4

How fast do factorials grow?

Accepted Answer

Factorials grow faster than exponentials. While 2^10 = 1,024, we have 10! = 3,628,800. At n = 20, the factorial has 19 digits. At n = 100, it has 158 digits. Stirling's approximation, n! is approximately sqrt(2 x pi x n) x (n/e)^n, helps estimate large factorials.

Question 5

What is the gamma function?

Accepted Answer

The gamma function extends factorials to non-integer and complex numbers. For positive integers, gamma(n) = (n-1)!. For example, gamma(5) = 4! = 24. It satisfies gamma(n+1) = n x gamma(n). Notably, gamma(1/2) = sqrt(pi), connecting factorials to pi. The gamma function is crucial in probability distributions and advanced mathematics.

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