Pumps a, b, and c operate at their respective constant rates. pumps a and b, operating simultaneously, can fill a certain tank in \frac{6}{5} hours; pumps a and c, operating simultaneously, can fill the tank in \frac{3}{2} hours; and pumps b and c, operating simultaneously, can fill the tank in 2 hours. how many hours does it take pumps a, b, and c, operating simultaneously, to fill the tank?

Let a, b, and c be the times each pump will fill the tank when working alone.

Therefore, in 1 hour;

1/a +1/b = 1/(6/5) = 5/6 (1)
1/a+1/c = 1/(3/2) = 2/3 (2)
1/b+1/c = 1/(2) = 1/2 (3)

From equation (1)
1/a = 5/6-1/b

Substituting for 1/a in eqn (2)
5/6-1/b+1/c = 2/3
-1/b +1/c = -1/6 => 1/c = 1/b - 1/6 --- (4)

Using eqn (4) in eqn (3)
1/b+1/b-1/6 = 1/2
2/b-1/6 = 1/2
2/b =1/2+1/6 = 2/3
1/b = 1/3
Then,
1/c = 1/3 - 1/6 = 1/6
1/a = 5/6 - 1/3 = 1/2

This means, in 1 hour and with all the pumps working together, the tank will be filled to;
1/a+1/b+1/c = 1/2+1/3+1/6 = 1 (filled fully).

Therefore, it will take 1 hour to fill the tank when all pumps are working together.

The question is kind of confusing. if it's asking how many variations of answers can you give. then do 4^5 for your answer.this would be something like the first question could be a and the 2nd c or first b next c etc. if you can, ask the teacher for a better confirmation of what it's asking. if i'm assuming correctly what it's asking, it's 625 different variations.