3
$\begingroup$

A certain school has $4$ forwards, $4$ guards, $3$ centers and $1$ person who can play as either a forward or a guard. How many different starting lineups can be made?

I came up with 2 answers to this problem. However I don't know which one is right and I can't tell the difference between the two:

Solution 1: There are two possibilities, X is a forward, in which there is $\binom{5}{2}\binom{4}{2}\binom{3}{1} = 180$ ways of making this starting lineup. X could be a guard as well, which results in the same number, $180$ ways of making the starting lineup. Add together to get $360$ different ways of making this starting lineup.

Solution 2: There are three possibilities which encompass all possible starting lineups: x is not picked, x is picked as a forward, and x is picked as a guard.

When x is not picked, there is $\binom{4}{2}\binom{4}{2}\binom{3}{1} = 108$ different lineups without x in it.

when x is picked as a forward, you only need to pick one more forward, so there is $\binom{4}{1}\binom{4}{2}\binom{3}{1} = 72$ different lineups with x as forward. the same number will result when you pick x as a guard: $72$. adding $108+72+72$ results in $252$ different lineups.

So the problem is that I can't see the fault in logic in either of my solutions. Which one is the right one?

edit: centers

$\endgroup$
4
$\begingroup$

The problem is that in solution $1$ you are counting the cases in which $X$ does not play twice, once when he doesn't play as a forward and once when he doesn't play as a guard. Therefore the answer should be $360$ minus the number of line-ups in which he doesn't play, this yields $360-108=252$ line-ups.

$\endgroup$
  • $\begingroup$ thanks, that makes sense. So my second solution is correct? $\endgroup$ – Gudushen Jan 30 '15 at 1:03
  • $\begingroup$ Yes. It is correct. $\endgroup$ – Jorge Fernández Hidalgo Jan 30 '15 at 1:05
  • $\begingroup$ @ModdedBear, from your answers, i see that you have a good grasp of combinatorics, counting problems and graph theory. i would like to get better at these. do you have any suggestions as how one could get better at these. $\endgroup$ – abel Jan 31 '15 at 18:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.