Probability combination Question

[J0EBL0W]

Homework Statement

We want to put 4 Maroon, 3 Green, and 2 Yellow plates in a row on a table. They are arranged randomly. What is the probability that the 4 Maroon plates are together?

Homework Equations

The Attempt at a Solution

I couldn't figure out how to set this problem up properly.

All I could think to do was to break it into the six outcomes we are interested in.

Let X denote a non-maroon plate and M denote a maroon plate.

So the outcomes are:
1- M M M M X X X X X
2- X M M M M X X X X
3- X X M M M M X X X
4- X X X M M M M X X
5- X X X X M M M M X
6- X X X X X M M M M

P(1) = (4/9)(3/7)(2/6)(1/5) = 1/126
P(2) = (5/9)(4/8)(3/7)(2/6)(1/5) = 1/126
P(3) = (5/9)(4/8)(4/7)(3/6)(2/5)(1/5) = 1/126
P(4) = ... = 1/126
P(5) = ... = 1/126
P(6) = ... = 1/126

So for a final answer I get 1/21. I'm not very confident in that answer, plus there has to be some way to set this up much simpler. Using the (n choose k) deal..

Any advice would be awesome.

 

[Dick]

Once you've picked a place to put the first maroon plate, all six patterns are equally likely (as you've probably noticed) (4/9)*(3/8)*(2/7)*(1/6)=1/126 (chance first maroon lies on selected place, time chance second maroon lies on second selected place etc). And you also correctly decided there are 6 equally probable ways to place them. I think you've done quite well. But you didn't need to write down a different product for each combination. Just IGNORE the non-maroon plates. They will sort themselves out.