Determine whether each situation involves a permutation or a...

Determine whether each situation involves a permutation or a combination. Then find the number of possibilities.
How many ways can a hand of five cards consisting of three cards from one suit and two cards from another suit be drawn from a standard deck of cards?

Key Concepts

Combination

A selection method in which the order of items does not matter. When choosing elements from a set without regard to arrangement, combinations are used to count the number of possible groups, typically calculated with binomial coefficients.

Binomial Coefficient

A mathematical expression, often noted as C(n, r) or 'n choose r', that represents the number of ways to choose r items from a collection of n distinct items without considering the order of selection. It is fundamental in solving problems involving combinations.

Multiplication Principle

A basic counting principle that states if one event can occur in m ways and another independent event can occur in n ways, then there are m x n ways for both events to occur together. This principle is used to combine independent counting scenarios in complex problems.

Transcript

00:03 Okay, so the number of ways that you could have five cards in a hand where three cards were from one suit and two cards are from another suit.

00:10 You really have to break this down quite a bit.

00:13 Those three cards from one suit, those could be three hearts or three diamonds or three spades or three clubs.

00:26 And then when we get all those numbers and that whole part figured out, we're going to multiply by because of the and.

00:34 Remember in counting when it's and you multiply by, the number of ways to get two cards from another suit.

00:40 Well, if you've already taken three cards from one suit, you only have three more suits to choose from.

00:45 So it could be three, or excuse me, it could be two cards from some suit that you haven't yet used, don't know what it is, or two cards from another suit you haven't yet used, whichever that is, or two cards from that other suit you haven't yet used, so there's only three options there because you already used one suit, you only have three suits left.

01:10 Now let's work the combinations in.

01:12 So there are 13 hearts available and if you're going to choose three, that would be 13 choose three.

01:18 For ore, when we count, we add.

01:21 There are 13 diamonds available.

01:23 You're choosing three.

01:24 So this is 13 choose three also.

01:26 Plus there are 13 spades.

01:29 You're choosing three.

01:30 So this is also 13 choose three...