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A sports coach intends to choose a team of players ...

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by gmattesttaker2 » Sun Aug 11, 2013 9:44 pm
Hello,

This is an IR question from MGMAT. Can you please assist with this?

A sports coach intends to choose a team of players from a pool of candidates. The coach wants to be able to have more than 20 but fewer than 25 distinct possibilities for the composition of the chosen team, with at least as many candidates chosen for the team as those not chosen.

Identify the number of candidates in the pool and the number of players on the team that are consistent with the coach's intentions.
Make only two selections, one in each column.

Number | Candidates in Pool | Players on Team
3
4
5
6
7
8

OA:
[spoiler]Candidates in Pool: 7
Players on Team: 5[/spoiler]
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by [email protected] » Sun Aug 11, 2013 11:07 pm
HI gmattesttaker2,

This question will require the Combination Formula (because you're forming possible teams and "order doesn't matter" in this particular situation).

This question requires a little bit of "brute force" to end up in the 20 < possible teams < 25 range, so you'd have to start quickly plugging in values until you find the right combo of numbers. There are some hints though, in that the number of players chosen has to be greater than or equal to the number not chosen, but the numbers have to be big enough and apart enough to give us the 21-24 possibilities that we're looking for.

As a "test case", what happens if you calculate 8c7? 8c6? 8c5? Notice how none of them fits what you're looking for? Look at the pattern that forms, then make the necessary adjustments to find the correct answer.

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by Brent@GMATPrepNow » Mon Aug 12, 2013 6:08 am
gmattesttaker2 wrote: A sports coach intends to choose a team of players from a pool of candidates. The coach wants to be able to have more than 20 but fewer than 25 distinct possibilities for the composition of the chosen team, with at least as many candidates chosen for the team as those not chosen.

Identify the number of candidates in the pool and the number of players on the team that are consistent with the coach's intentions.
Make only two selections, one in each column.

Number | Candidates in Pool | Players on Team
3
4
5
6
7
8
As Rich mentioned, this question requires some brute force. It also requires us to be able to quickly calculate combinations (preferably in our head).

Aside: we have a free video on calculating combinations in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

So, let's start with 8 candidates and check various numbers of players on the team. We'll calculate 8C7, 8C6, 8C5 and 8C4 (we'll stop there, because the question tells us that at least as many candidates chosen for the team as those not chosen.

IMPORTANT: When calculating combinations (using the fast method described in our video), it's useful to apply the following rule: nCr = nC(n-r). In other words, "n choose r" is equal to "n choose n - r"
So, for example: 10C7 = 10C3
12C9 = 12C3
8C7 = 8C1
etc.

Okay, now let's start testing cases.
8C7 = 8C1 = 8. 8 possible teams is too small (we want between 20 and 25 teams)
8C6 = 8C2 = 28. 28 possible teams is too big (we want between 20 and 25 teams)
8C5 = 8C3 = 56. 56 possible teams is too big (we want between 20 and 25 teams)
Since 8C4 will definitely be bigger than 56, we can rule this out as well.

Now, we'll try combinations where we begin with 7 candidates.
7C6 = 7C1 = 7. 7 possible teams is too small (we want between 20 and 25 teams)
7C5 = 7C2 = 21. PERFECT!

Answer: 7 Candidates and 5 players on the team

Cheers,
Brent
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by gmattesttaker2 » Tue Aug 13, 2013 8:53 am
Brent@GMATPrepNow wrote:
gmattesttaker2 wrote: A sports coach intends to choose a team of players from a pool of candidates. The coach wants to be able to have more than 20 but fewer than 25 distinct possibilities for the composition of the chosen team, with at least as many candidates chosen for the team as those not chosen.

Identify the number of candidates in the pool and the number of players on the team that are consistent with the coach's intentions.
Make only two selections, one in each column.

Number | Candidates in Pool | Players on Team
3
4
5
6
7
8
As Rich mentioned, this question requires some brute force. It also requires us to be able to quickly calculate combinations (preferably in our head).

Aside: we have a free video on calculating combinations in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

So, let's start with 8 candidates and check various numbers of players on the team. We'll calculate 8C7, 8C6, 8C5 and 8C4 (we'll stop there, because the question tells us that at least as many candidates chosen for the team as those not chosen.

IMPORTANT: When calculating combinations (using the fast method described in our video), it's useful to apply the following rule: nCr = nC(n-r). In other words, "n choose r" is equal to "n choose n - r"
So, for example: 10C7 = 10C3
12C9 = 12C3
8C7 = 8C1
etc.

Okay, now let's start testing cases.
8C7 = 8C1 = 8. 8 possible teams is too small (we want between 20 and 25 teams)
8C6 = 8C2 = 28. 28 possible teams is too big (we want between 20 and 25 teams)
8C5 = 8C3 = 56. 56 possible teams is too big (we want between 20 and 25 teams)
Since 8C4 will definitely be bigger than 56, we can rule this out as well.

Now, we'll try combinations where we begin with 7 candidates.
7C6 = 7C1 = 7. 7 possible teams is too small (we want between 20 and 25 teams)
7C5 = 7C2 = 21. PERFECT!

Answer: 7 Candidates and 5 players on the team

Cheers,
Brent

Hello Brent,

Thank you very much for the detailed explanation and for the excellent video. Your explanation makes things so much easier to understand. Thanks again for all your help.

Best Regards,
Sri
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