Can anyone help me with this simple math problem?

Question

Some students in a gym class are wearing blue jerseys, and the rest are wearing red jerseys. There areexactly 25 ways to pick a team of three players that includes at least one player wearing each color. Compute the number of students in the class.

Answer 1

Nice problem.



If you have both colors in the group of 3,

it's either (2 blue + 1 red), or (2 red + 1 blue).



Let b = number of blues, and r = number of reds.



Number of ways to the first is bC2 * r.

Number of ways to do the second is rC2 * b



bC2 = b(b-1)/2

rC2 = r(r-1)/2



So br (b-1) / 2 + br (r-1) / 2 = 25.

br (b - 1 + r - 1) = 50

br (b + r - 2) = 50



But everything has to be integers (no fractional kids, you know).

So these numbers must be factors of 50.

50 = 5 * 5 * 2



So there are two possibilities:

1) both colors have 5 kids, but then b + r - 2 = 8, not 2, so that doesn't work.



2) One color has 5 kids and the other has 2, then 5 + 2 - 2 = 5,

and we have 5 * 2 * 5 and it works.





There are 7 kids in the class, divided 5 and 2 between the colors, but we don't know which way.





If the first word, "Some", were replaced by "Most" or "More than half" or some other quantifier, it would say which color had more kids, and disambiguate the problem.



Checking:

Let's say 5 blues and 2 reds.

A 2-blue-1-red team can be chosen in 5*4 * 2 / 2 = 20 ways

A 2-red-1-blue team can be chosen in 1 * 5 = 5 ways.

Total is 25.

Done.



P.S. Thanks for all the Thumbs Ups, guys! I appreciate it.

Answer 2

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Answer 3

Well, it is easy. there are 75 students. 25 x 3 = 75.