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A class is divided into four groups of four students each.

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A class is divided into four groups of four students each. [#permalink]

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New post 02 Feb 2012, 11:07
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A class is divided into four groups of four students each. If a project is to be assigned to a team of three students, none of which can be from the same group, what is the greatest number of distinct teams to which the project could be assigned?

(A) 4^3
(B) 4^4
(C) 4^5
(D) 6(4^4)
(E) 4(3^6)

My solution:

4C3 * 4 = 4^4 .
Is the formula used rightly selected although the answer agrees to OA?
[Reveal] Spoiler: OA
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Re: Selecting a team from four groups [#permalink]

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New post 02 Feb 2012, 11:22
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docabuzar wrote:
Q. A class is divided into four groups of four students each. If a project is to be assigned to a team of three students, none of which can be from the same group, what is the greatest number of distinct teams to which the project could be assigned?

(A) 4^3
(B) 4^4
(C) 4^5
(D) 6(4^4)
(E) 4(3^6)

My solution:

4C3 * 4 = 4^4 .
Is the formula used rightly selected although the answer agrees to OA?


Basically we are asked to determine the # of different teams of 3 that can be formed so that no 2 members are from the same group.

\(C^3_4=4\) - choosing which 3 groups out of 4 will provide with a member;
\(C^1_4*C^1_4*C^1_4=4^3\) - choosing each member from the selected 3 groups ;

So, total # of different groups will be: \(4*4^3=4^4\).

Answer: B.

P.S. Answer is correct, formula and math is not: \(C^3_4*4=4^2\).
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Re: A class is divided into four groups of four students each. [#permalink]

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New post 02 Feb 2012, 11:34
Thanks.

Its clear now. I m making the silly mistake of reading my 4*4 as 4^4 from the OA.
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Re: A class is divided into four groups of four students each. [#permalink]

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Re: A class is divided into four groups of four students each. [#permalink]

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New post 04 Sep 2016, 22:28
The problem can be solved in relatively simpler way without using combinations formula but by simply using counting method.

We have to fill 3 places by selecting 3 students from 4 groups of 4 such that no student is from same group.
Lets create groups:
1st group - a,b,c,d
2nd group - e,f,g,h
3rd group - i,j,k,l
4th group - m,n,o,p
The first position can be filled in 16 ways (a,b....p), 2nd can be filled in 12 ways - since 2nd student should not be from the same group (e,f....p), 3rd position can be filled in 8 ways (i,j,....p)
So number of ways they can be arranged is 16*12*8 = 4*4*4*3*4*2
This gives the number of arrangements where the order is important, but for us order is not important - It does not matter to us whether we select 'a' first of 'g' second... All that is important to us is that we have to select 3 people.
So we have to un-arrange the number of arrangements. We can do so by dividing the total number of arrangements with 3!.
So the final answer is 4*4*4*3*4*2/3*2 = 4^4.

Hence (B).
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Re: A class is divided into four groups of four students each.   [#permalink] 04 Sep 2016, 22:28
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