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In a certain group of 10 developers, 4 developers code only in Python

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In a certain group of 10 developers, 4 developers code only in Python [#permalink]

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In a certain group of 10 developers, 4 developers code only in Python and the rest program in either Ruby on Rails or PHP - but not both. If a developer organization is to choose a 3-member team, which must have at least 1 developer who codes in Python, how many different programming teams can be chosen?

(A) 100
(B) 40
(C) 66
(D) 80
(E) 75
Spoiler: OA

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Re: In a certain group of 10 developers, 4 developers code only in Python [#permalink]

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Bunuel wrote:
In a certain group of 10 developers, 4 developers code only in Python and the rest program in either Ruby on Rails or PHP - but not both. If a developer organization is to choose a 3-member team, which must have at least 1 developer who codes in Python, how many different programming teams can be chosen?

(A) 100
(B) 40
(C) 66
(D) 80
(E) 75


Two ways...
1) total ways = 10C3 = 10!/7!3! = 120..
ways without python developer = 6C3= 6!/3!3!=20..
ways of atleast ONE python developer = 120-20 =100..

2) ways of selecting ONLY one = 4*6C2 = 4*15 = 60..
ways of selecting ONLY two = 4C2*6C1 = 6*6 = 36..
ways of selecting all three = 4C3 = 4 = 4..
Total = 60+36+4=100...
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Re: In a certain group of 10 developers, 4 developers code only in Python [#permalink]

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New post 27 May 2016, 01:52
There are 4 python developers.

Whenever there is a case of "at least one is to be chosen", it is always better to use (Total - When none is chosen)

When there are no restrictions, we can select 3 members out of 10 people, in 10C3 ways: This works out to 120.

Now when we have to select 3 persons out of 10 persons such that NONE of them is a Python developer, then, this is as good as selecting 3 persons out of 6 persons (leave out all 4 python developers). This can be done in 6C3 ways = 20 ways.

120-20 = 100 ways. These are the different ways in which we can choose a team such that at least one Python developer is included.
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Re: In a certain group of 10 developers, 4 developers code only in Python [#permalink]

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New post 27 May 2016, 02:10
Bunuel wrote:
In a certain group of 10 developers, 4 developers code only in Python and the rest program in either Ruby on Rails or PHP - but not both. If a developer organization is to choose a 3-member team, which must have at least 1 developer who codes in Python, how many different programming teams can be chosen?

(A) 100
(B) 40
(C) 66
(D) 80
(E) 75


Number of 3-member programming teams that can be formed in which at least one member of the group codes in Python = 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 codes in Python
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 codes in Python = 6!/3!3! = 20
Number of different teams of 3 that can be formed in which at least one member of the group codes in Python = 120 - 20 = 100
Answer - A
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Re: In a certain group of 10 developers, 4 developers code only in Python [#permalink]

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New post 08 Feb 2018, 16:50
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Bunuel wrote:
In a certain group of 10 developers, 4 developers code only in Python and the rest program in either Ruby on Rails or PHP - but not both. If a developer organization is to choose a 3-member team, which must have at least 1 developer who codes in Python, how many different programming teams can be chosen?

(A) 100
(B) 40
(C) 66
(D) 80
(E) 75


We can use the formula:

Number of ways to select at least 1 developer who codes in Python = total number of ways - Number of ways to select no developers who code in Python

Total number of ways to select the group:

10C3 = 10!/[3! (10-3)!] = 10!/[3! 7!] = (10 x 9 x 8)/(3!) = (10 x 9 x 8)/(3 x 2 x 1) = 10 x 3 x 4 = 120 ways

Total number of ways to select no developers who code in Python:

6C3 = 6!/[3! (6-3)!](6 x 5 x 4)/(3!) = (6 x 5 x 4)/(3 x 2 x 1) = 20

Number of ways to select at least 1 developer who codes in Python = 120 - 20 = 100

Answer: A
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Re: In a certain group of 10 developers, 4 developers code only in Python   [#permalink] 08 Feb 2018, 16:50
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