A manufacturer shows 40 bathing suits to a buyer. Use equation (9.20) to determine how many ways

A manufacturer shows 40 bathing suits to a buyer. Use...

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and selecting objects, often under specific constraints. It provides the tools and formulas necessary to calculate the number of ways to choose items from larger sets, which is essential in solving problems where order does not matter.

Combination Formula

The combination formula, often written as 'n choose k', is used to determine the number of ways to select k items from a set of n items without regard to the order of selection. This formula is given by n!/(k!(n-k)!), where n! (n factorial) represents the product of all positive integers up to n.

Factorial

Factorial notation, denoted by n!, represents the product of all positive integers from 1 to n. It plays a crucial role in the combination formula and other areas of combinatorics, as it helps simplify the expressions used to calculate permutations and combinations.

Transcript

00:01 Let's talk about the difference between a combination or permutation.

00:04 What's the difference here? well, combination, order does not matter.

00:09 So here, if i were to select events a, b, and c, this is the same as selecting events b, a, than c, because the same three were selected, doesn't matter which order i chose them from.

00:20 Whereas a permutation order does, in fact, matter.

00:25 So selecting events a, b, and c in that order is different than events b, a, than c.

00:31 Okay, so in this case, let's say a buyer wants to stock eight different posters.

00:37 How many ways can they, she select eight if she has 20 to choose from? does it matter what order she stocks eight different posters? no, she just wants to buy them.

00:47 So this will be a combination.

00:49 So it's going to be, i have 20 and i'm choosing eight.

00:53 So this formula is the first number factorial over the bottom number factorial, over subtract is two numbers factorial.

01:01 So those two numbers, subtracted is going to be 12 factorial approximately.

01:10 So this will be 20 factorials, 2019, times 18, 17, 16, 15, 14, 15, 14 times 13 times 12 factorial.

01:20 Sorry, right out of space.

01:21 12 factorial on bottom, then i have 8 factorial, 8, 7, 6, 5, 4, times 3, times 2, and times 1.

01:29 The 12, what's going to cancel here? the 12 factorials will cancel.

01:34 Okay, what else can cancel? eight times two is 16, so that can cancel...

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