What are factorials and how are they used in combinatorics?

What are factorials and how are they used in combinatorics?

Factorials are the product of all positive integers up to a given number. In combinatorics, they help calculate permutations and combinations.

Can you give me an example of how to calculate a factorial?

Sure! For example, 5! (5 factorial) is 5 × 4 × 3 × 2 × 1 = 120.

How do factorials relate to combinations?

Factorials are used in the formula for combinations: C(n, k) = n! / (k! * (n-k)!). This calculates how many ways you can choose k items from n.

What about permutations? How do they use factorials?

Permutations also use factorials, calculated as P(n, k) = n! / (n-k)!. This gives the number of ways to arrange k items from n.

Are there any special properties of factorials I should know?

Yes! For instance, n! grows very quickly, and 0! is defined as 1. Factorials are also used in series expansions.