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The University of Maryland, University of Vermont, and Emory University have each $4$ soccer players. If a team of $9$ is to be formed with an equal number of players from each university, how many number of ways can the selection be done?

Possible answers are: $3, 4, 12, 16, 25$.

I have encountered that problem in GRE Quant.

Here is my solution: From each university we should take $9:3=3$ players and this choice could be done with $C_4^3=4$ ways. Hence the selection can be done in $4\times 4 \times 4=4^3=64$ ways. But such answer does not exist.

Can anyone explain this moment please?

YuiTo Cheng
  • 4,705
RFZ
  • 16,814

2 Answers2

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I'd frame exactly the same computation you use as "the number of ways to pick the one player that is excluded from each University ...". Of course, I'd get the same answer...

As a quick enumeration of at least 26 different ways to make some of these teams...

Let $a, b, c, d$ represent the four players from the University of Maryland. Let $f, g, h, i$ represent the four players from the University of Vermont. Let $p, q, r, s$ represent the four players from Emory University. The question is this ungrammatical mess "If a team of 9 is to be formed with an equal number of players from each university, how many number of ways can the selection be done?" In particular, "how many number of ways" is not grammatically correct English. Nevertheless, we will represent a "selection" by who from each University is not selected (since listing the three selected players is equivalent to listing the one non-selected player from each University). \begin{align*} afp && afq && afr && afs \\ agp && agq && agr && ags \\ ahp && ahq && ahr && ahs \\ aip && aiq && air && ais \\ \\ bfp && bfq && bfr && bfs \\ bgp && bgq && bgr && bgs \\ bhp && bhq && &&\text{+2 more} \\ &&&&\text{+4 more} \\ \\ &&&&\text{+16 more} \\ \\ &&&&\text{+16 more} \\ \\ \end{align*}

Eric Towers
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I cross check with other answer is 12=4+4+4

The question is asking selection for each university hence Principal of Addition will apply. Principal of Multiplication will be applied when asked about number of possible combination of team. The question specifically asks selection from each university not selection of team

Samar Imam Zaidi
  • 8,800
  • Can you explicitly illustrate these 12 selections? – RFZ Aug 25 '17 at 05:38
  • This question is tricky and test Principal of Addition/Multiplucation Concept, GRE has purposefully removed 64 so that people will ponder over this question – Samar Imam Zaidi Aug 25 '17 at 05:40
  • The answer 12 is not persuasive. I would like to look at these 12 choices in some certain example. – RFZ Aug 25 '17 at 05:41
  • It is testing "Principal of Addition" vrs "Principal of Multiplication" concept. Number of Sekection from Each university=4 total selection =12, but if asked number of team formed it is 444=64 – Samar Imam Zaidi Aug 25 '17 at 05:45
  • Still can not understand you. Could you explain it in real example? – RFZ Aug 25 '17 at 05:49
  • If there are two jobs such that they can be performed independently then principal of Addition is used. I am quoting one example – Samar Imam Zaidi Aug 25 '17 at 05:56
  • Teacher has to select either 1 boys from 20 boys or one girl from 20 girls hence number of ways=20+20 – Samar Imam Zaidi Aug 25 '17 at 05:58
  • Thanks a lot for example. However this did not clarify to me that initial problem. It asks in how many way the selection can be done? Why we are using rule of sum? Little bit ambiguous. – RFZ Aug 25 '17 at 06:16
  • Yes I agree but GRE asks question of this type "how many number of ways can the selection be done?" He is not asking how many ways team can be formed.Read Mutually Exclusive or Disjoint event, it will be clear – Samar Imam Zaidi Aug 25 '17 at 06:33
  • @SamarImamZaidi : Your answer contradicts your comment to the OP. Also, the overwhemlingly important word in your 20 boys or 20 girls example is "or", which means that you only select one child. In the given problem, you must select one player from U. Maryland and one player from U. Vermont and one player from Emory U. to form a team. – Eric Towers Aug 25 '17 at 12:44