Blue Book: Math, pg. 657, #15

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1

<p>Guys, I need help with this question. I just can't figure out how the answer can be 24.</p>

<p>
[quote]
15. The Acme Plumbing Company will send a team of 3 plumbers to work on a certain job. The company has 4 experienced plumbers and 4 trainees. If a team consists of 1 experienced plumber and 2 trainees, how many different such teams are possible?

[/quote]
</p>

<p>"If a team consists of 1 experienced plumber and 2 trainees, how many different such teams are possible?"</p>

<p>So, that would mean 4 plumbers x 4 trainees x 3 trainees (1 picked already), which would mean 48. Now, how is it 24 and not 48?</p>

2

<p>The order doesn't matter in this question, so you're doing it wrong by using permutation for the trainees--permutation counts the order.</p>

<p>You have to use combination... 4C1 x 4C2 = 4 x 4x3/2 = 24</p>

<p>In your method, exp + trainee + trainee and trainee + trainee + exp count as different teams, but they're the same.</p>

3

<p>Wow, I just realized that. Thanks for the explanation. I have to start being more careful on this stuff.</p>

4

<p>Yeah, this kind of stuff confuses me as well.</p>

<p>Just to make sure though, how would one calculate this if the order mattered?</p>

<p>So 24 teams are possible.... then you do permutation for 3?</p>

<p>24 x 3! which equals 144. Can anyone verify this?</p>

5

<p>Since order does not matter, because one plumber is the same as another and one trainee is the same as another trainee, you first</p>

<p>Find all possible combinations of 2 trainees out of 4 possible , so in calc
4 nCr 2 = 6
Then you need one out of 4 plumbers at a time, so there are 4 possibilities
4 nCr 1 = 4</p>

<p>6 x 4 = 24</p>

6

<p>I did it the long way didnt take me that long..</p>

7

<ol>
<li>The Acme Plumbing Company will send a team of 3 plumbers to work on a certain job. The company has 4 experienced plumbers and 4 trainees. If a team consists of 1 experienced plumber and 2 trainees, how many different such teams are possible?</li>
</ol>

<p>Best way to approach this:</p>

<p>Fact 1: There are 4 different experienced plumbers.
Fact 2: There are 6 different combinations of 2 trainees in a total group of 4 trainees. If this isn't obvious, let the 4 trainees be A,B,C,and D. </p>

<p>All the possible trainee pairs:</p>

<p>AB,AC,AD
BC,BD
CD</p>

<p>and each of these trainee pairs can go with any of the 4 experienced plumbers, so we multiply 6 X 4 for the total number of possible teams, which gives 24. A common error here would be to erroneously consider AB a different pair from BA, when really order doesn't matter in this problem. That is how you would get a wrong answer of 48.</p>