Question

In how many ways can 3 prizes be distributed among 4 boys, when

(i) no boy gets more than one prize?
(ii) a boy may get any number of prizes?
(iii) no boy gets all the prizes?

Answer 1

(i) We have  to distribute  3  prizes  in 4 boys. if no any  boy   gets  more than a  one  prize than we have to select   3 boys.

Number  of ways  to select  3  boys   = ⁴C₃

Also We can  arrange  3  prizes  in 3! ways

Hence, Total number  of  ways  = ⁴C₃ × 3! = 24

(ii) Now, in  second  a  boy can  get  any  number  of  prize. We  can  give  fisrt  prize to any  of  4  boys , second  prize  to any  of  4  boys and also third  to any  of  4  boys .

Total number  of  ways = 4 × 4 × 4 = 64

(ii) if  no boys get  all prizes than , we have  to remove  those  ways  from 64  in which a  boy  get  all prizes .  Clearly,  we can given all prizes  to first  or  second  or  third  or  fourth boy .

Hence,

Total number  of  ways  = 64  - 4 = 60

Answer 2

The correct answer to this questio is Option (ii) A boy may get any number of prizes?

This is possible since one boy may be able to win two prizes in a competition.

However, the rest of the options, namely ''no boy gets more than one prize?'' or ''no boy gets all the prizes? ''

Would not really work, as with the first option, we still might have a left over prize, while in the second option, no one will get it.