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Task force teams exercise

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Task force teams exercise

by Lucas C » Wed Aug 17, 2016 12:52 pm
A certain city contains 25 consulates, each of which employs 25 people. If an international research task force of 24 people is to be chosen from these people, none of whom can come from the same consulate, what is the greatest number of distinct task force teams that could be formed?

a) 24^25
b) 25^24
c) 25^25
d) 24(25^24)
e) 25(24^25)

Answer: C



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I don't understand the problem with this reasoning:

I can't pick more than 1 from the same consulate -> I must choose 1 from each consulate.
25C1 = 25
Since I have to do the same for every consulate, that gives me 25^25 combinations of people that could possibly be on the same team, i. e., teams of 25 people to choose from.

For every team of 25 people, I must pick 24 of them. So:
25C24=25

Given the number of combinations, I can have 25^25 different teams to pick from * 25 different picks for each team.

So 25^26 was my answer.
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by Brent@GMATPrepNow » Wed Aug 17, 2016 1:59 pm
Lucas C wrote:A certain city contains 25 consulates, each of which employs 25 people. If an international research task force of 24 people is to be chosen from these people, none of whom can come from the same consulate, what is the greatest number of distinct task force teams that could be formed?

a) 24^25
b) 25^24
c) 25^25
d) 24(25^24)
e) 25(24^25)

Answer: C
Take the task of creating the international research task force and break it into stages.

Stage 1: Select the 24 consulates (from which we will select 1 employee from each)
Since the order in which we select the consulates does not matter, we can use combinations.
We can select 24 consulates from the 25 consulates women in 25C24 ways (25 ways)
So, we can complete stage 1 in 25 ways

NOTE: this is the same as selecting 1 consulate to NOT have an employee chosen. That is, 25C1 = 25C24 = 25

Stage 2: From 1 of the chosen consulates, select 1 employee
Since there are 25 employees in a consulate, we can complete this stage in 25 ways.

Stage 3: From 1 of the REMAINING consulates chosen, select 1 employee
Since there are 25 employees in a consulate, we can complete this stage in 25 ways.

Stage 4: From 1 of the REMAINING consulates chosen, select 1 employee
Since there are 25 employees in a consulate, we can complete this stage in 25 ways.

Stage 5: From 1 of the REMAINING consulates chosen, select 1 employee
Since there are 25 employees in a consulate, we can complete this stage in 25 ways.

.
.
.
.
Since we choose 24 consulates in stage 1, we will have to make 24 employee selections (1 from each chosen consulate).
.
.
.

Stage 24: From 1 of the REMAINING consulates chosen, select 1 employee
Since there are 25 employees in a consulate, we can complete this stage in 25 ways.

Stage 25: From the last REMAINING consulate chosen, select 1 employee
Since there are 25 employees in a consulate, we can complete this stage in 25 ways.

By the Fundamental Counting Principle (FCP), we can complete all 25 stages (and thus create the create the international research task force) in (25)(25)(25)(25)(25)(25)(25)(25)(25)(25)(25)(25)(25)(25)(25)(25)(25)(25)(25)(25)(25)(25)(25)(25)(25) ways ([spoiler]= 25^25 ways[/spoiler])

Answer: C
--------------------------

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by Matt@VeritasPrep » Fri Aug 19, 2016 2:07 am
Here's how I'd think of it.

Since you're picking 24 of the 25 consulates, start by choosing the one you DON'T want. There are 25 options for this.

Now, from each of the 24 consulates you DID want, you have 25 choices of representative. So you've got

(25 choices of consulate to leave out) * 25²� (25 people at each of 24 consulates)

25 * 25²� gives us C, so we're good to go (and spoiled for choice ...)
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