There are 3 math clubs in the school district, with 5, 7, and 8 students respectively. Each club has two

Sorry, Full question: There are 3 math clubs in the school district, with 5, 7, and 8 students respectively. Each club has two co-presidents. If I randomly select a club, and then randomly select three members of that club to give a copy of a book, what is the probability that two of the people who receive books are co-presidents?

Can anyone help me with this probability problem? Thanks so much! :D

using the law of total probability,
Pr = (1/3)(2c2*3c1/5c3 + 2c2*5c1/7c3 + 2c2*6c1/8c3)
= (1/3)(3/10 + 5/35 + 6/56) = 11/60 , ≈ 0.1833 <------

prob to select a club = 1/3
---------------------
1 club with 5 students
number of combinations of 3 = 5C3 = 5!/(3! x 2!) = 10
number of combinations of 3 with the 2 copresidents
= 5-2 = 3
prob to select the 2copresidents among the 3 members selected
= 3/10

2 club with 7 students
number of combinations of 3 = 7C3 = 7!/(3! x 4!) = 35
number of combinations of 3 with the 2 copresidents
= 7-2 = 5
prob to select the 2copresidents among the 3 members selected
= 5/35

3 club with 8 students
number of combinations of 3 = 8C3 = 8!/(3! x 5!) = 56
number of combinations of 3 with the 2 copresidents
= 8-2 = 6
prob to select the 2copresidents among the 3 members selected
= 6/56

answer to find :

(1/3 x 3/10) + (1/3 x 5/35) + (1/3 x 6/56) =