In a certain group of 10 developers, 4 developers code...

LUANDATO wrote:In a certain group of 10 developers, 4 developers code only in Python and the rest program in either Ruby or Rails or PHP - but not both. If a developer organization is to choose a 3-members 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

The OA is A.

The team needs at least 1 Python coder, meaning 1 , 2 or 3

Start with 1 Python coder. 4 ways to choose. Now need to pick 2 coders from the other 6. Two slots available. First slot can be filled 6 ways, second slot 5 ways. However, this assumes order matters, which it doesn't, so need to divide by two:

4 x (6x5)/2 = 60 ways

Next select 2 Python Coders. First slot can be filled 4 ways, second slot 3 ways, Remember to divide by 2 for the reason mentioned above. Now fill non-Python coder slot, 6 ways:

(4 x 3)/2 x 6 = 36 ways

Finally, fill all three slots with Python coders. First slot can be filled 4 ways, second slot 3 ways and third slot 2 ways. Need to divide by 3 x 2 x 1 to eliminate permutations of the same basic group, e.g. ABC is the same as CAB, etc.

(4 x3 x 2)/(3 x 2 x 1) = 4 ways

The combination formula is just a simplified way to eliminate permutations of the same members where order doesn't matter.

Add up all the ways: 60 + 36 + 4 =100, A

LUANDATO wrote:In a certain group of 10 developers, 4 developers code only in Python and the rest program in either Ruby or Rails or PHP - but not both. If a developer organization is to choose a 3-members 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