a) You need to use the formula for the number of ways of choosing r items/people from n items/people

`C(n,r) = (n!)/(r!(n-r)!)`

We say 'n choose r'

Here `n=17` and `r=3` so the number of ways is

`C(17,3) = (17!)/(3!14!) = (17.16.15.14.13...3.2.1)/((3.2.1)(14.13.12...3.2.1))=(17.16.15)/(3.2.1)`

Cancelling on the top and bottom gives `C(17,3) = 17.8.5 = 680`  

b) Now we have `C(17,3)` ways of choosing 3 individuals each receiving 1 prize, plus `C(17,2)` ways of choosing 2 individuals receiving 2 prizes and 1 prize respectively, plus `C(17,2)` ways of choosing 2 individuals receiving 1 prize and 2 prizes respectively, plus `C(17,1)` ways of choosing 1 individual receiving all 3 prizes.

So we have the number of ways is

`C(17,3) +2C(17,2) + C(17,1) = 680 + 2((17!)/(2!15!)) + (17!)/(1!16!)`

` = 680 + 2((17.16.15.14.13...3.2.1)/((2.1)(15.14.13...3.2.1))) + (17.16.15.14.13...3.2.1)/((1)(16.15.14.13...3.2.1))`

` ``= 680 + 2((17.16)/(2.1)) + (17)/(1) = 680 + 2.17.8 + 17 = 680 + 272 + 17 = 969`

a) 680 ways

b) 969 ways