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?

(i) First prize can be given to any one of the 4 boys, so it can be given in 4 ways. Second prize can be given in 3 ways, as the boy getting first prize can not get the second. Similarly, third prize can be given to the remaining 2 boys in 2 ways.

Hence, the total number of ways in which no boy gets more than one prize = 4 × 3 × 2 = 24

(ii) First prize can be given to any one of the 4 boys, so it can be given in 4 ways. Second prize can also be given in 4 ways, as the boy getting first prize can also get the second. Similarly, third prize can be given in 4 ways.

Hence, the total number of ways in which a boy can get any number of prizes = 4 × 4 × 4 = 64

(iii) Number of ways in which a boy gets all the 3 prizes is 4, as any one of the 4 boys may get all the prizes.

Hence, the total number of ways in which a boy does not get all the prizes = 64 – 4 = 60

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