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# There are 5 students and 3 teachers. In how many ways can a team of 5

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Math Expert
Joined: 02 Sep 2009
Posts: 62353
There are 5 students and 3 teachers. In how many ways can a team of 5  [#permalink]

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13 Feb 2020, 06:27
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Difficulty:

45% (medium)

Question Stats:

66% (01:33) correct 34% (02:13) wrong based on 29 sessions

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There are 5 students and 3 teachers. In how many ways can a team of 5 be formed so that there is at least one teacher but not more than two teachers in the team?

A. 450
B. 180
C. 60
D. 45
E. 30

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Posts: 2012
Re: There are 5 students and 3 teachers. In how many ways can a team of 5  [#permalink]

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13 Feb 2020, 07:54
We have two ways to make a suitable team, so we have two cases, and we'll need to add the numbers from each case. We can either:

- use 1 teacher and 4 students. Then we have 3 choices for the teacher, and 5 choices for the students (since we're just leaving one student out), and, multiplying our choices, we have 3*5 = 15 possible teams

- use 2 teachers and 3 students. Then we have 3 choices for the teachers (since we're just picking which teacher to leave out), and 5C3 = (5)(4)/2 = 10 choices for the two students, since we're picking 2 from 5 and order does not matter. Multiplying our choices, we can make 3*10 = 30 such teams.

Adding the results from each case, we have 15+30 = 45 possible teams in total.
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Joined: 22 May 2019
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There are 5 students and 3 teachers. In how many ways can a team of 5  [#permalink]

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13 Feb 2020, 12:43
students = 5
teachers = 3
total = 8

team capacity = 5
condition = teacher must be (atleast 1 and not more than 2)
possible ways : T S S S S OR T T S S S = 3C1*5C4 + 3C2*5C3 => 3*5+3*10 => 45 ways//
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Re: There are 5 students and 3 teachers. In how many ways can a team of 5  [#permalink]

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14 Feb 2020, 07:37
total possible ways
1 teacher & 4 student to form a team of 5 ; 3c1*5c4 +3c2*5c3 ; 15+30 ; 45
IMO D

Bunuel wrote:
There are 5 students and 3 teachers. In how many ways can a team of 5 be formed so that there is at least one teacher but not more than two teachers in the team?

A. 450
B. 180
C. 60
D. 45
E. 30
Re: There are 5 students and 3 teachers. In how many ways can a team of 5   [#permalink] 14 Feb 2020, 07:37
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