(i). We have to find the possible number of ways in which we can give four prizes among five students when no boy gets more than one price which means that there is no repetition.

We will use the concept of multiplication because there are four sub jobs dependent on each other and are performed one after the other.

The thing that is distributed is considered to have choices not the things to which we have to give them, it means that in this problem the prizes have choices more precisely first prize will have five choices, second prize will have four choices and the choices will keep on decreasing by one as we go on giving prizes, and students won’t choose any because prizes will have the right to choose.

The number of ways in which we can give four prizes among five students where repetition of distribution is not allowed 5 × 4 × 3 × 2 = 5! = 120.

(ii). We have to find the possible number of ways in which we can give four prizes among five students when any student can get any number of prices which means that there is a repetition of prizes.

We will use the concept of multiplication because there are four sub jobs dependent on each other and are performed one after the other.

The thing that is distributed is considered to have choices, not the things to which we have to give them; it means that in this problem the prizes have choices more precisely five choices are there for each prize and students won’t choose any because prizes have the right to choose.

The number of ways in which we can give four prizes among five students where repetition of distribution is allowed 5 × 5 × 5 × 5 = 5⁴ = 625.

(iii). We have to find the possible number of ways in which we can give four prizes among five students when no student gets all the prizes.

We will use the concept of multiplication because there are four sub jobs dependent on each other and are performed one after the other.

The thing that is distributed is considered to have choices, not the things to which we have to give them; it means that in this problem the prizes have choices more precisely five choices are there for each prize and students won’t choose any because prizes have the right to choose.

The number of ways in which we can give four prizes among five students when no student gets all the prices

= number of ways in which any student can get any number of prices – the number of ways in which one student gets all the prizes.

= 64 – 5 = 59.

There are five students, so any one student will get all the prizes hence there are five choices to give all the prizes to any one student.