We roll five standard dice (sides numbered 1 to 6) and write down the sum of the top three i.e. of the 3 highest values.
What is the probability to get 15 ?
Total possible outcomes = 6^5
15 can be made with 6,6,3 or 6,5,4 or 5,5,5
6,6,3 can be made in 130 ways.
(6,6,3,2,1 has 5!/2! = 60 ways
6,6,3,1,1 and 6,6,3,2,2 have 5!/2!2! = 30 ways each
6,6,3,3,1 and 6,6,3,3,2 have 5!/2!2! = 30 ways each
6,6,3,3,3 has 5!/2!3! = 10 ways)
Similarly
6,5,4 can be made in 740 ways
654XY 360 ways
654XX 180 ways
6544X 180 ways
65444 20 ways
5,5,5 can be made in 181 ways.
555XY 120 ways
5555X 20 ways
555XX 40 ways
55555 1 way
all can be done in 1051 ways
probability is 1051 / 6^5 = .135159
analytical attempt at it
