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A firm is divided into four departments, each of which contains four [#permalink]

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21 Sep 2009, 09:21

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A firm is divided into four departments, each of which contains four people. If a project is to be assigned to a team of three people, none of which can be from the same department, what is the greatest number of distinct teams to which the project could be assigned?

Re: A firm is divided into four departments, each of which contains four [#permalink]

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21 Sep 2009, 10:08

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my approach

4 Dept (I,II,III,IV) with 4 people in each dept.

To choose 1st team member = we have 16 options (4 dept x 4 people) To choose 2nd member = we have 12 options (3 dept x 4 people) To choose 3rd member = we have 8 options ( 2 dept x 4 people)

unique teams = 16 x 12 x 8...i don't know where i am wrong ..help me ...

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A firm is divided into four departments, each of which contains four people. If a project is to be assigned to a team of three people, none of which can be from the same department, 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)

First, we need to choose the three teams we'll take employees from. That's the same as choosing one team *not* to take an employee from, so we can choose our three teams in 4 ways. Then we have 4 choices for which employee we take from each of the three teams, so the answer is 4*4*4*4 = 4^4.

bhushan252 wrote:

my approach

4 Dept (I,II,III,IV) with 4 people in each dept.

To choose 1st team member = we have 16 options (4 dept x 4 people) To choose 2nd member = we have 12 options (3 dept x 4 people) To choose 3rd member = we have 8 options ( 2 dept x 4 people)

unique teams = 16 x 12 x 8...i don't know where i am wrong ..help me ...

You're assuming that there is a '1st team member', a '2nd team member' and a '3rd team member' - that is, you're assuming the order of the team members is somehow important. If the question asked in how many ways we might choose a President, Vice-President and Treasurer with the given restrictions, your answer would be correct. However, since the order of our 3 team members doesn't matter, you need to divide your answer by 3! = 6. That gives (16)(12)(8)/(6) = 16*2*8 = 2^8 = 4^4.
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Re: A firm is divided into four departments, each of which contains four [#permalink]

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21 Sep 2009, 20:24

IanStewart wrote:

powerka wrote:

A firm is divided into four departments, each of which contains four people. If a project is to be assigned to a team of three people, none of which can be from the same department, 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)

First, we need to choose the three teams we'll take employees from. That's the same as choosing one team *not* to take an employee from, so we can choose our three teams in 4 ways. Then we have 4 choices for which employee we take from each of the three teams, so the answer is 4*4*4*4 = 4^4.

bhushan252 wrote:

my approach

4 Dept (I,II,III,IV) with 4 people in each dept.

To choose 1st team member = we have 16 options (4 dept x 4 people) To choose 2nd member = we have 12 options (3 dept x 4 people) To choose 3rd member = we have 8 options ( 2 dept x 4 people)

unique teams = 16 x 12 x 8...i don't know where i am wrong ..help me ...

You're assuming that there is a '1st team member', a '2nd team member' and a '3rd team member' - that is, you're assuming the order of the team members is somehow important. If the question asked in how many ways we might choose a President, Vice-President and Treasurer with the given restrictions, your answer would be correct. However, since the order of our 3 team members doesn't matter, you need to divide your answer by 3! = 6. That gives (16)(12)(8)/(6) = 16*2*8 = 2^8 = 4^4.

Re: A firm is divided into four departments, each of which contains four [#permalink]

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12 Jul 2010, 16:56

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@bhushan252 YOu calcualted the permutation, in which we consider a team (r, s, d) and (d,s,r) different. However they are same when it comes about selecting people. SO divide you permutation result by r! Permutation: n!/(n-r)! Combintaiton: permutation/r! so your result that is 16 x 12 x 8 divided by 3! is the answer that is 4^4 (B). Cheers!
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Re: A firm is divided into four departments, each of which contains four [#permalink]

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13 Jul 2010, 05:30

Guys, I got the right answer, but i am not sure if my approach is right. Lets say we have only 3 groups from which we have to pick one person from each group. The the number of combinations would be 4*4*4. Now, since we have 4 groups and we need to pick three at a time, the number of ways we can do this is 4C1=4. Hence total number of ways u can pick 3 people (1 person from each grp) from four groups is 4 (4*4*4) = 4 ^ 4.

What if we had six grps, wud the answer be 6C3 * 4*4 *4 ? GMAT experts, please help.

Guys, I got the right answer, but i am not sure if my approach is right. Lets say we have only 3 groups from which we have to pick one person from each group. The the number of combinations would be 4*4*4. Now, since we have 4 groups and we need to pick three at a time, the number of ways we can do this is 4C1=4. Hence total number of ways u can pick 3 people (1 person from each grp) from four groups is 4 (4*4*4) = 4 ^ 4.

What if we had six grps, wud the answer be 6C3 * 4*4 *4 ? GMAT experts, please help.

Correct. If there were 6 departments then: \(C^3_6\) # of ways to choose which 3 department will provide employee for the team and as each chosen department can provide with 4 employees then total # of different teams will be \(C^3_6*4*4*4=5*4^4\).

A firm is divided into four departments, each of which contains four people. If a project is to be assigned to a team of three people, none of which can be from the same department, 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)

The same approach for the original question: \(C^3_4\) # of ways to choose which 3 department will provide employee for the team and as each chosen department can provide with 4 employees then total # of different teams will be \(C^3_4*4*4*4=4^4\).

Re: A firm is divided into four departments, each of which contains four [#permalink]

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31 Aug 2011, 09:34

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klueless7825 wrote:

I'm trying to solve the problem using reverse combination approach.Please,Can anyone help.

Need to choose 3 people from 16 so 16C3.

No. of ways to choose 3 people from the same team 4C3.since there are four teams (4C3)^4

16C3 - (4C3)^4

=560-256

=304

First of all, this approach is complicated, convoluted and error prone. I'd go by Bunuel's explanation. Different permutation and combination problems are done using different methods. You just need to pick the most apt one for a specific type.

Anyway, No. of ways to choose 3 people from the same team 4C3.since there are four teams (4C3)^4

This is not entirely correct. Number of ways to choose 3 people from the same team: 4C3. Correct There are four teams: (4C3)^4. Not correct.

Remember, here you need to add the numbers not multiply because you choose from 1 OR choose from 2 OR choose from 3 OR choose from 4.

Secondly, you have missed out another case in which just 1 candidate is picked from 1 team AND 2 candidates are picked from another team. Because, that will also be the NOT one candidate from each team.

So, actual answer could be arrived by this: \(Total=C^{16}_{3}=560\)

Picking all 3 candidates from just one team: \(C^4_1*C^4_3=16\) OR(i.e. +) Picking 2 candidates from one team AND picking 1 candidate from one of the remaining teams \(C^4_1*C^4_2*C^3_1*C^4_1=288\)

Total number of ways in which 1 candidate is NOT picked from each team \(16+288=304\)

Now, to find the left-overs just subtract from total: \(560-304=256=4^4\)

Re: A firm is divided into four departments, each of which contains four [#permalink]

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30 Sep 2011, 01:33

Bunuel wrote:

anandnat wrote:

Guys, I got the right answer, but i am not sure if my approach is right. Lets say we have only 3 groups from which we have to pick one person from each group. The the number of combinations would be 4*4*4. Now, since we have 4 groups and we need to pick three at a time, the number of ways we can do this is 4C1=4. Hence total number of ways u can pick 3 people (1 person from each grp) from four groups is 4 (4*4*4) = 4 ^ 4.

What if we had six grps, wud the answer be 6C3 * 4*4 *4 ? GMAT experts, please help.

Correct. If there were 6 departments then: \(C^3_6\) # of ways to choose which 3 department will provide employee for the team and as each chosen department can provide with 4 employees then total # of different teams will be \(C^3_6*4*4*4=5*4^4\).

A firm is divided into four departments, each of which contains four people. If a project is to be assigned to a team of three people, none of which can be from the same department, 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)

The same approach for the original question: \(C^3_4\) # of ways to choose which 3 department will provide employee for the team and as each chosen department can provide with 4 employees then total # of different teams will be \(C^3_4*4*4*4=4^4\).

Answer: C.

Hope it helps.

Thanks Bunuel for the clear explanation. I got it.

Re: A firm is divided into four departments, each of which contains four [#permalink]

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12 Nov 2013, 06:23

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Re: A firm is divided into four departments, each of which contains four [#permalink]

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12 Nov 2013, 08:12

This is the method that I followed.

Method 1:

You need to fill in three spots

_ x _ x _

from these guys Group-1 (a,b,c,d) Group-2 (e,f,g,h) Group-3 (i,j,k,l) Group-4 (m,n,o,p)

Since each member of the final 3 member group must be different, you choose one from each group (group-1,2,3 or 4). Now you have a choice of 4 people to select from, so for the group-1

4 x _ x _

group-2 again gives you four choices, so

4 x 4 x _

group-3 again gives you four choices

4 x 4 x 4

The same can be done using different combo of 4 groups. That combo is equal to 4C3 which is again equal to 4. Multiply that as well

Re: A firm is divided into four departments, each of which contains four [#permalink]

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30 Dec 2013, 08:00

powerka wrote:

A firm is divided into four departments, each of which contains four people. If a project is to be assigned to a team of three people, none of which can be from the same department, 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)

Source: Jeff Sackmann's GMAT Extreme Challenge

We have four departments with 4 people in each

Now we need to pick 3 people of different departments, let's do it step by step

First, pick 3 departments among 4 different options Hence 4C3 = 4

Now, pick between for people in each department Hence, total combo is 4*4*4

Finally we then have 4^4 for total available options

B is the best choice here

Hope it helps! Cheers! J

PS. Does anybody have some more questions from Source: Jeff Sackmann's GMAT Extreme Challenge?

Re: A firm is divided into four departments, each of which contains four [#permalink]

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08 Dec 2015, 11:49

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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