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25 May 2016, 04:22
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
35%
(medium)
Question Stats:
72% (01:58) correct
28%
(02:07)
wrong
based on 464
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A certain team consists of 4 professors and 6 teaching assistants. How many different teams of 3 can be formed in which at least one member of the group is a professor? (Two groups are considered different if at least one group member is different.)
A. 48
B. 100
C. 120
D. 288
E. 600
A. 48
B. 100
C. 120
D. 288
E. 600
04 Jun 2018, 10:47
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Bunuel
When we see a counting question involving "at least", we should consider using the nice rule:
Total number of outcomes that FOLLOW a rule = (TOTAL number of outcomes that IGNORE the rule) - (number of outcomes that BREAK the rule)
Here, we have: Total number of teams with AT LEAST one professor = (TOTAL number of teams with ANY NUMBER of professors) - (number of teams with ZERO professors)
TOTAL number of teams with ANY NUMBER of professors
W'ere ignoring the rule that talks about the number of professors on a team.
There are 10 people in total, and we must select 3 to be on a team.
Since the order in which we select the people does not matter, we can use COMBINATIONS
We can select 3 people from 10 people in 10C3 ways (= 120 teams)
Number of teams with ZERO professors
This means all 3 team members must be assistants
There are 6 assistants in total, and we must select 3 of them to be on a team.
Since the order in which we select the assistants does not matter, we can use COMBINATIONS
We can select 3 assistants from 6 assistants in 6C3 ways (= 20 teams)
ASIDE: If anyone is interested, we have a free video (below) on calculating combinations (like 6C3) in your head
Total number of teams with AT LEAST one professor = (TOTAL number of teams with ANY NUMBER of professors) - (number of teams with ZERO professors)
= 120 - 20
= 100
Answer: B
Cheers,
Brent
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25 May 2016, 05:36
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The groups can be formed as
PTT + PPT + PPP where P=professor and T=Teaching assistant
Number of different teams of 3 can be formed in which at least one member of the group is a professor
= 4C1* 6C2 + 4C2*6C1 + 4C3
= 4!/3! * 6*5/2 + 4*3/2 * 6 + 4
=60 +36 + 4
=100
Answer B
PTT + PPT + PPP where P=professor and T=Teaching assistant
Number of different teams of 3 can be formed in which at least one member of the group is a professor
= 4C1* 6C2 + 4C2*6C1 + 4C3
= 4!/3! * 6*5/2 + 4*3/2 * 6 + 4
=60 +36 + 4
=100
Answer B
