Solve the following permutation/ combination question

Question

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?

Answer 1

Many of these problems are based on some similar concepts.  Unlike your other problem, this one actually is a permutation.  The reason is because these are positions on the baseball team.  If you kind of think of 9 slots to fill, if order didn't matter we could move the people around in those slots and you'd still have the same set of 9 people.  But since the slots are specific positions on the team (like pitcher, catcher, right-fielder, etc), then it's going to make a difference if they're moved around.  Hence, the order matters.

 

So this is a permutation of picking 9 out of 12.

 

If you need the permutation equation, you can do  P(12,9) =   12!

                                                                                                          (12-9)!

 

(Again, if you're having problems solving those, let us know.  Being able to set them up is different than solving.  If you have trouble solving the factorial equations, we need to make sure we go over that.)

 

The other way to see this is that for the first slot you have 12 choices, and with one down (you can't repeat), you now only have 11 choices, and then 10 choices, etc, and you go until you've picked 9 total:

12*11*10*9*8*7*6*5*4

(Stop there - that's 9. This is your multiplication rule.)

 

And that is the same as P(12,9) - it's just the permutation equation is a shortcut.  It's a bit different way of thinking about it.

 

(EDIT: Now fixed for my error so you can ignore my comment below.  I was trying to pick 12 and kept thinking "isn't a baseball team 9?"  Oh my!)