In how many ways can 3 prizes be distributed among 4 boys, when
(i) no boy gets more than 1 prize;
(ii) a boy may get any number of prizes;
(iii) no boy gets all the prizes?
In how many ways can 3 prizes be distributed among 4 boys, when
(i) no boy gets more than 1 prize;
(ii) a boy may get any number of prizes;
(iii) no boy gets all the prizes?
Case (i) When no boy gets more than 1 prize.
The first prize may be given to any of the 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 = (4×3×2)=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 = (4×4×4)=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.
Case (i) When no boy gets more than 1 prize.
The first prize may be given to any of the 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 = (4×3×2)=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 = (4×4×4)=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.
