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
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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.

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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