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A certain team consists of 4 professors and 6 teaching assistants. How
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25 May 2016, 04:22
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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.)
Re: A certain team consists of 4 professors and 6 teaching assistants. How
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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
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Re: A certain team consists of 4 professors and 6 teaching assistants. How
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26 May 2016, 09:14
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Bunuel wrote:
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
At least one professor= Total possibility- All teaching assistants
Total possibility= 10C3= 120 Teaching assistant= 6C3= 20
AT least one professor= 120-20= 100
B is the answer
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Re: A certain team consists of 4 professors and 6 teaching assistants. How
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26 May 2016, 10:03
Bunuel wrote:
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
Number of different teams of 3 that can be formed in which at least one member of the group is a professor = Total Number of different teams of 3 that can be formed - Number of different teams of 3 that can be formed in which no member of the group is a professor Total Number of different teams of 3 that can be formed = 10!/7!3! = 120 Number of different teams of 3 that can be formed in which no member of the group is a professor = 6!/3!3! = 20 Number of different teams of 3 that can be formed in which at least one member of the group is a professor = 120 - 20 = 100 Answer - B
A certain team consists of 4 professors and 6 teaching assistants. How
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04 Jun 2018, 10:47
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Bunuel wrote:
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
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