probability problem? how many ways could sitters be arranged

I am stuck on this:

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?

The answer is 24 but I can't figure out how they arrived to that answer.
Is this a probability question?

Re: probability problem?

It's a problem in permutations. If Martha is in the middle, how many ways can the other 4 sit around her?

Re: probability 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?

The answer is 24 but I can't figure out how they arrived to that answer.
Is this a probability question?

Click to expand...

Do you know why there are 120 different ways they can sit? It’s the same as asking how many different orders can five things be arranged in. The answer is 5! (5x4x3x2x1).

Once Martha’s position is assigned to the middle, the problem is changed; now there are just four objects (people) to be arranged. The number of ways they can be arranged is 4!

4! = 24

Make sense?

For this to be a probability problem, it would have been worded differently, such as, “What is the probability that Martha sits in the middle if the friends all take their seats randomly?”. The answer to this would be the number of ways/times that Martha ends up in the middle (24) divided by the total number of ways the friends could arrange themselves (120). The probability that Martha ends up in the middle is

24/120 = .2

So you can see that Combinations and/or Permutations are sometimes useful in finding probabilities.

Re: probability problem?

Draw a sketch.

__ __ M __ __

How many choices are there for the 1st blank (seat) to be occupied by a friend? 4.
Once that seat is filled, how many choices for the second seat? 3.
etc.
When you have each of these answers, multiply them and what do you get?

Re: probability problem?

thanks. I see it now. For those wondering, these are practice problems from the ETS's GRE study guide. It's been over 10 years since I took statistics and math. Highest level of math I took was Calc I.
You guys are awesome.
studying by yourself from a book, I find myself missing the classroom where there is feedback.
thanks again!