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A team of five people is to be formed from a pool of 5 professionals

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Bunuel
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A team of five people is to be formed from a pool of 5 professionals [#permalink] New post   29 Sep 2022, 01:14
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A team of five people is to be formed from a pool of 5 professionals and 3 amateurs. The number of teams that can be formed such that the team contains at least 3 professionals is how many fewer than the total number of teams that can be formed (that is, teams formed with no mandatory conditions)?

(A) 10
(B) 11
(C) 26
(D) 46
(E) 56
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A
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SaquibHGMATWhiz
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Re: A team of five people is to be formed from a pool of 5 professionals [#permalink] New post   29 Sep 2022, 04:28
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Bunuel wrote:
A team of five people is to be formed from a pool of 5 professionals and 3 amateurs. The number of teams that can be formed such that the team contains at least 3 professionals is how many fewer than the total number of teams that can be formed (that is, teams formed with no mandatory conditions)?

(A) 10
(B) 11
(C) 26
(D) 46
(E) 56

Solution:

  • Total number of teams that can be formed = Choosing 5 from 8 choices \(= 8C5=56\)

  • Total number of teams containing at least 3 professionals \(=30+15+1=46 \)
    • 3 professionals AND 2 amateurs \(=5C3\times 3C2=10\times 3=30\)
    • 4 professionals AND 1 amateur \(=5C4\times 3C1=5\times 3=15\)
    • 5 professionals \(=5C5=1\)

  • Thus, \(56-46=10\)


Hence the right answer is Option A
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Rabab36
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Re: A team of five people is to be formed from a pool of 5 professionals [#permalink] New post   29 Sep 2022, 04:57
no of ways in which 5 team can be formed without any condition is

8!/5!3!= 56 ways

now we have to form team containing atleast 3 professionals
lets find team with 3 amateurs and 2 professionals

let us say AAAPP
i.e 5!/3!2! = 10 ways

therefore team containing atleast 3 professionals
total ways - team with 3 amateurs and 2 professionals

56-10= 46 ways


now according to question
56-46= 10 ways
therefore correct option A
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Re: A team of five people is to be formed from a pool of 5 professionals [#permalink] New post   30 Sep 2022, 10:39
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Bunuel wrote:
A team of five people is to be formed from a pool of 5 professionals and 3 amateurs. The number of teams that can be formed such that the team contains at least 3 professionals is how many fewer than the total number of teams that can be formed (that is, teams formed with no mandatory conditions)?

(A) 10
(B) 11
(C) 26
(D) 46
(E) 56



{Total - (At least 3)} MEANS cases where we have less than 3 professionals.

Now, we have 3 amateurs, so we require 5-3 or 2 professionals.

Only combination is (All 3 amateurs) + (2 out of 5 professionals) = 3C3*5C2 = 1*10 = 10 ways.

A
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Re: A team of five people is to be formed from a pool of 5 professionals [#permalink]
30 Sep 2022, 10:39
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