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/
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