# Counting and combinations

1. Sep 17, 2016

### gtfitzpatrick

1. The problem statement, all variables and given/known data
A club has only 8 women and 6 men as members. A team of 3 is to be chosen to represent the club. In how many ways can this be done if there is to be at least one woman on the team.

2. Relevant equations

3. The attempt at a solution

I can do this 2 ways,
first 1w2m + 2w1m + 3m0m => (8c1)(6c2) + (8c2)(6c1) + (8c3)(6c0) =344
or
total- all men => (14c3) - (6c3) = 344

But the problem i cant get my head around is why this way doesnt work: (8c1)(13c2)
can some one please throw some light on this for me

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2. Sep 17, 2016

### andrewkirk

Because you are putting a partial order on the the three people selected. You select one, and then you select a pair to make up three. So some sets of three are counted twice. For instance, the case where you first pick Laxmi, then pick Mae and Brunhilde (together) will be treated as different from the one where you first pick Mae, then Laxmi and Brunhilde (together). But as a combination, they are just one instance, not two.

3. Sep 17, 2016