David R. answered • 04/05/14

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I LOVE THIS QUESTION! I use it in my math class as a way to illustrate the difference between combinations and permutations while also requiring students to use BOTH in the same problem.

Start with a smaller example: Imagine for you have 4 people named BOB, MARY, SAM, and TONY. How many batting orders can be created with only 3 people?

Well first you have to select which 3 people to use. First, select which 3 people are on the team (which is a combination of any 3 people). AFTER you create the team, find how many ways to order the team into a line-up.

- 4C3 = total number of teams that can be made
- 3P3 = total number of ways to arrange 3 people into a line-up***

***Notice that 3P3 is equivalent to 3! which means (3*2*1)= 6.*

***4C3 is equivalent to 4!/(3!1!) = 6*In combinatorics, it is good to remember that the words "OR" and "AND" mean to "add" and "multiply" respectively.

Since the process says "build a team AND THEN order it" this means that the number of teams made needs to be multiplied by the number of ways the team can be arranged into the line-up:

- 4C3 * 3P3 = 6 * 6 = 36 total possible line-ups

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For question 2:

- Since 1 game has 9 innings this means 1 game = 9 innings
- Each inning is a new line-up which means 1 inning = 1 line-up. Putting these two lines together means 1 game = 9 line-ups.
- 195 games would mean (9*195) line-ups needed for the entire season.

Use the answer for question 1 to help you out to see if there's enough line-up to go through the season without repeating OR how many games could be played before you would HAVE TO repeat a line-up.