Combinatorics Solution Explanation

[aaronhma]

Pretty standard combinatorics question:

Jenny has $3$ skirts, $2$ pants, and $4$ shirts. How many ways can she create an outfit with one of the shirts, and either a skirt or pant?

Can someone please explain why the solution is $20$? Thanks!

[西瓜]

Hello.
She has 4 shirts to pick from. She can pick either a skirt or a pant and there are 3+2 = 5 of those total. So, there are five ways she can pick a skirt or pant.

4x5 =20 ways total.

[audrey]

@西瓜 Great explanation! Adding on to that, this is something that's formally known as the "Multiplication Principle." It basically means that if you can break a task into two independent events (meaning the events don't depend on each other), and one event can happen in x ways, and another can happen in y ways, then the whole task can occur in xy ways. That might sound sort of convoluted, but in practice, it's more intuitive

For example, if a meal consists of a main dish and a drink, and there are 3 main dishes and 2 drinks, there are 3 * 2 = 6 total meals! (There are other ways to double check this-- drawing a tree diagram or just listing them out are two possibilities)

Or, in your problem, one "event" is choosing a shirt, and the other is choosing a bottom (skirts or pants). There are 4 ways to choose a shirt and 3+2 = 5 ways to choose a bottom, so there are 4 * 5 = 20 ways to choose an "outfit" (top and bottom).

[aaronhma]

@西瓜 @audrey Thank you both for the awesome explanations