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solve the following permutations questions

1- There are nine different positions on a baseball team. If a team has 12 players how many different line-ups can the team make?
 
2-Baseball games consist of nine innings. A team wants to change its line-up every inning. If no game goes to extra innings, and a season consists of 195 games, how many complete seasons can the team play without repeating a line-up?
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1 Answer

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
This same logic could be use to answer question 1.
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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.