# Math Help - Problem - distinguishable permuations

1. ## Problem - distinguishable permuations

I am currently taking a course in Calculus, and a course in Probability/Statistics at my local University. Probability has always been my weak spot in math, so I might post questions here from time to time. Especially since we are stuck with "Probability and Statistical Inference" by Hogg and Tanis as our textbook. This book has no student solution's manual, so it can be a very frustrating book to work with.

Anyway, here goes:

There are three teams in a cross-country race. Team A has five runners, team B has six runners, and team C has seven runners. In how many ways can the runners cross the finish line if we are interested only in the team for which they run? That is, what is the number of distinguishable permutations of five A's, six B's, and seven C's? (Note that, for scoring purposes, only the scores of the five first runners for each team count.)

Any tips on solving this problem will be greatly, greatly appreciated!

2. Originally Posted by krje1980
There are three teams in a cross-country race. Team A has five runners, team B has six runners, and team C has seven runners. In how many ways can the runners cross the finish line if we are interested only in the team for which they run? That is, what is the number of distinguishable permutations of five A's, six B's, and seven C's? (Note that, for scoring purposes, only the scores of the five first runners for each team count.)
I am not at all sure that I understand that question.

If it means "How many ways can the string AAAAABBBBBBCCCCCCC be arranged"?
The answer is $\frac{18!}{5!\cdot 6!\cdot 7!}$.

If it means "How many ways can the string AAAAABBBBBCCCCC be arranged"?
The answer is $\frac{15!}{(5!)^3}$.

If it means neither of those, then what does it mean?

3. Hi.

Thanks for your suggestions. I had actually tried both of your suggestions before I submitted the question here, but none of them give the correct answer (which according to the book is 14702688).

I agree with you in that the question sounds confusing, but that is exactly how it is written. I think it sucks that our professor decided to use this book as it has gotten quite a lot of negative feedback on amazon exactly because it is often obtuse and unclear.

4. Note that I corrected a typo in my answer.
$\frac{18!}{5!\cdot 6!\cdot 7!}=14702688$.

5. Oh yeah, I see it now! Great! Thank you! I kept trying to get the problem to work out with 15! in the numerator - I guess the way the problem was presented confused me. Now I understand! I really appreciate it