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An engagement team consists of a project manager, team leade

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Sonia2023

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25 Aug 2023, 05:58

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Bunuel

Sonia0106

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 out of 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

EMPOWERgmatRichC Bunuel chetan2u

Can anyone explain what is wrong with this approach:

a) No of ways to select consultants when 1 of the consultant is not included:

2C1 * 3C1 * 6C4 = This gives me 90 ways in which 1 consultant out of the 2 is not selected.

b) No of ways to select consultants when the second consultant is not included:

2C1 * 3C1 * 6C4 = This gives me 90 ways.

So total number of ways are 180

Can someone explain which case I am missing because of which the difference if 30 (180-150)?

If one of the consultants is not included, then another one is included, so we need to select the remaining 3 out of 5: 5C3 = 10. Hence, for this case, we'd have 2C1*3C1*5C3 = 60. The situation is the same when the other consultant is not included: 2C1*3C1*5C3 = 60.

Finally, we should consider the number of committees where neither of those two consultants is included, which means that we need to select 4 out of 5 remaining consultants. In this case, we'd have 2C1*3C1*5C4 = 30.

60 + 60 + 30 = 150.

Answer: C.

Thanks Bunuel for a quick reply. Post reading your reply, I realized what mistake I made while considering my initial cases. The folks other than the 2 consultants are being double counted when I am taking 6C4, so I need to remove the groups of 4 consultants which are common in CASE 1 and CASE 2 - which can be done in 5C4 ways (5 remaining consultant and 4 to be selected from them) = 5 ways

Then 2C1*3C1*5C4 = 30 cases need to be removed from 180.

Really appreciate your quick response.

Thank you.

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