In how many ways can 3 prizes be distributed among 4 boys, when
(i) no boy gets more than 1 prizes ,
(ii) a boy may get any number of prizes,
(iii) no boy gets all the prizes?

Explanation

Case (i) When no boy gets more than 1 prize.
The first prize may be given to any of hte 4 boys in 4 ways.
The 2nd prize may be given in 3 ways, since the boy who got the 1st prize cannot receive it.
The 3rd prize may be given to any of the remaining 2 boys in 2 ways.
Required number of ways`=(4xx3xx2)=24.`
Case (ii) When a boy may get any number of prizes.
The 1st prize may be given to any of the 4 boys in 4 ways.
Since a boy may get any number of prizes, so 2nd prize may be given to any of the 4 boys in 4 ways.
Similarly, the 3rd prize may be given in 4 ways.
Required number of ways `=(4xx4xx4)=64.`
Case (iii) When no boy gets all the prizes.
The number of ways in which a boy gets all the prizes is 4, as anyone of the 4 boys may get all the prizes.
Hence, the number of ways in which a boy does not get all the prizes `=(64-4)=60.`