# Thread: Factorials in a word problem

1. ## Factorials in a word problem

Here is the word problem:

Martha invited 4 friends to go with her to the movies. There are 120 different ways in which they can sit together in a row. In how many of those ways is Martha sitting in the middle?

I need help with setting up the problem.
If you can help, I really appreciate it!

Here is the word problem:

Martha invited 4 friends to go with her to the movies. There are 120 different ways in which they can sit together in a row.
Because n objects can be arranged in a row in $\displaystyle n!=1\cdot 2\cdot 3\cdots n$ ways.

In how many of those ways is Martha sitting in the middle?

I need help with setting up the problem.
If you can help, I really appreciate it!
Just imagine yourself placing these 5 people on 5 the seats of a row: You first place Martha in the middle (no alternative choice is possible here). The only choice that you have is how to arrange the 4 friends on the remaining 4 seats. And in how many ways can that be done? (Note that it doesn't matter whether the 4 remaining seats are next to each other or not.)

Here is the word problem:

Martha invited 4 friends to go with her to the movies. There are 120 different ways in which they can sit together in a row. In how many of those ways is Martha sitting in the middle?

I need help with setting up the problem.
If you can help, I really appreciate it!
_ _ Martha _ _

The first seat can be filled in 4 ways, the second seat in 3 ways, the third seat in only one way (Martha), the fourth seat in 2 ways and the fifth seat in one way.

There are 4x3x1x2x1=24 ways.

So, Martha is sitting in the middle in 24 of those ways.