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) The first prize can be given to any of the four boys. Then, the second prize can be given to any of the three boys. Lastly, the third prize can be given to any one of the remaining 2 boys. 

∴ The number of ways in which all the 3 prizes can be given = 4 × 3 × 2 = 24. 

(ii) In this case, each of the three prizes can be given in 4 ways since a boy can receive any number of prizes. 

∴ The number of ways in which all the prizes can be given = 4 × 4 × 4 = 4³ = 64. 

(iii) Since anyone of the 4 boys can get all the prizes, therefore, the number of ways in which a boy gets all the 3 prizes = 4. 

∴ Number of ways in which a boy does not get all the prizes = 64 – 4 = 60.