Rock, Paper, Scissors

Date: 03/29/2001 at 00:06:28
From: Jennifer Baum
Subject: Combinations/permutations

Hi,

My class is studying for a standardized exam and one of the questions 
was this:

If three people are playing Rock Paper Scissors, how many different 
combinations can be made, assuming the order doesn't matter? 

With this example, our instructor said it was easier just to write 
them down such as RRR, RRP, RRS, RPP, RPS, RSS, PPP, PPS, PSS, SSS, 
and the answer is obviously 10. Although that gets us through the 
exam, I was still wondering how you would manage if there were 5 
people with 5 choices.

It seems so simple. What am I missing?

Date: 03/29/2001 at 00:57:43
From: Doctor Schwa
Subject: Re: Combinations/permutations
>It seems so simple.  What am I missing?

Hi Jennifer,

This is actually a pretty tricky type of problem. The usual formulas 
and patterns for permutations or combinations don't apply without a 
little extra work. I learned this technique just a few years ago 
myself, when co-coaching a high school math team.

The method I learned is to think of it as dividing the people into 
three groups, the rock group, the paper group, and the scissors group. 
I don't care which people go into which group; just how many people 
end up in each group.

One way to represent that is with xxx standing for the three people 
and | | as the separators between groups.

Then, for instance, your RRR would look like:   xxx |    |
     and your RSS looks like:                    x  |    | xx
     and your PPS looks like:                       | xx | x

Does that make sense?

Then the question becomes: how many ways are there to put five objects 
in order if three of them are identical x's (the people) and two of 
them are identical separators (the |'s)?

The answer to that is 5 choose 2, or in general, if you have n people 
and k groups (k different symbols), then it's (n+k-1) choose (k-1).

Thanks for an interesting question!

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/