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# An engagement team consists of a project manager, team leader, and

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An engagement team consists of a project manager, team leader, and  [#permalink]

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06 Jun 2017, 11:32
1
7
00:00

Difficulty:

55% (hard)

Question Stats:

66% (02:39) correct 34% (02:39) wrong based on 109 sessions

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An engagement team consists of a project manager, team leader, and four consultants. There are 2 candidates for the position of project manager, 3 candidates for the position of team leader, and 7 candidates for the 4 consultant slots. If 2 of the 7 consultants refuse to be on the same team, how many different teams are possible?

A. 25
B. 35
C. 150
D. 210
E. 300

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An engagement team consists of a project manager, team leader, and  [#permalink]

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Updated on: 13 Jun 2017, 16:09
1
Project manager - 2
Four consultants - 7

Total number of combinations = 2$$C_1$$ * 3$$C_1$$ * 7$$C_4$$ = 2*3*35 = 210

Total number of combinations when both consultants are together = 2$$C_1$$ * 3$$C_1$$ * 5$$C_2$$ = 2*3*10=60

Number of Teams possible = 210-60 = 150

Originally posted by quantumliner on 06 Jun 2017, 12:34.
Last edited by quantumliner on 13 Jun 2017, 16:09, edited 1 time in total.
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Re: An engagement team consists of a project manager, team leader, and  [#permalink]

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13 Jun 2017, 14:58
Quote:
Total number of combinations when both consultants are together

Should total up to 60 vice 40.
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Re: An engagement team consists of a project manager, team leader, and  [#permalink]

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13 Jun 2017, 16:48
1
C
The number of ways to choose the project manager is 2 and the number of ways to choose a team leader is 3.
The total number of ways to choose 4 consultants out of 7 candidates is 7!/(4!3)= 35, however, we must also take into the consideration that 2 candidates refuse to work together and for that reason, we must deduct from the total number of ways all the options when these 2 are together. We do it by counting all the outcomes, when these 2 are already in team and there are still 2 spots available: 5!/(2!*3!) = 10, so there are 35-10=25 ways of picking 4 candidates out of 7, taking into the consideration the fact that 2 of them don't want to work together. The only thing that remains now is to multiply all outcomes 2*3*25=150
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Re: An engagement team consists of a project manager, team leader, and  [#permalink]

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02 Aug 2017, 13:25
quantumliner wrote:
Project manager - 2
Four consultants - 7

Total number of combinations = 2$$C_1$$ * 3$$C_1$$ * 7$$C_4$$ = 2*3*35 = 210

Total number of combinations when both consultants are together = 2$$C_1$$ * 3$$C_1$$ * 5$$C_2$$ = 2*3*10=60

Number of Teams possible = 210-60 = 150

How do we get 5C2 when we calculate the total number of combinations when consultants are together?
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Re: An engagement team consists of a project manager, team leader, and  [#permalink]

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03 Aug 2017, 06:08
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TheMastermind wrote:
How do we get 5C2 when we calculate the total number of combinations when consultants are together?

Hi TheMastermind ,

When I am saying two consultants are always together, that means I already have them on my bucket and now I have to select the pending 2 from the remaining 5 consultants. Thus, we are using 5C2

Let me know if you need the detailed explanation for this question.
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Re: An engagement team consists of a project manager, team leader, and  [#permalink]

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18 Apr 2019, 23:06
I understand the explanations given above. I want to understand where i m going wrong with my approach.

So, the team need to have 1 PM, 1 TL and 4 Cs. We have 2 PMs, 3 TLs and 7Cs. Of the 7Cs, 2 wont work together.

# of ways in which 1 PM can be chosen: 2 [2C1]

# of ways in which 1 TL can be chosen: 3 [3c1]

# of ways 4 Cs can be chosen: # of ways 5 Cs with no issues can be chosen + # of ways 6 Cs (5 Cs with no issues + 1 of the 2 Cs with issues) can be chosen * 2 (since either 1 of the 2 Cs having issues can be chosen)

--> 5C4 + 6C4*2 = 35

I understand that 35 is also 7C4. However, I am not able to understand how 5C4 + 6C4*2 includes the cases in which the 2 consultants having issues are put in the same team.

Bunuel, It would be very helpful if you could point out where the flaw in my reasoning is. Thanks in advance
Re: An engagement team consists of a project manager, team leader, and   [#permalink] 18 Apr 2019, 23:06
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