In how many ways can three persons, each throwing a single dice once, make a sum of 15?

Explanation

Number on the faces of the dice are 1,2,3,4,5,6 (least number 1, greatest number 6)
here, l=1, m=6, r=3 and n=15
`therefore`Required number of ways=Coefficient of `x^(15-1xx3)` in the expansion of `(1-x^(6))^(3)(1-x)^(-3)`
=Coefficient of `x^(12)` in the expansion of
`(1-3x^(6)+3x^(12))(1+.^(3)C_(1)x+.^(4)C_(2)x^(2)+ . . .+.^(8)C_(6)x^(6)+ . ..+.^(14)C_(12)x^(12)+ . . .)`
`=.^(14)C_(12)-3xx.^(8)C_(6)+3=.^(14)C_(2)xx.^(8)C_(2)+3`
`=91-84+3=10`