Question

There are nine students ($5$ boys and $4$ girls) in the class. In how many ways:
(i) One student (either girl or boy) can be selected to represent the class.
(ii) A team of two students (one girl and one boy) can be selected.
(iii) Two medals can be distributed (no one get both).
(iv) One prize for Maths, two prizes for Physics and three prizes for Chemistry can be distributed (No student can get more than one prize in same subject and prizes are distinct).

Answer 1

(i)

There are $9$ students, of whom one needs to be selected.

This is given by :

${}^9{C}_1=\frac{9!}{(9-1)!1!}=9$

(ii)

One girl has to be selected from a group of 4, while one boy should be selected from among 5, in order to form the team. This is given by:

${}^4{C}_1\times {}^5{C}_1=4\times 5=20$

(iii)

In order to distribute 2 medals among 9 students, we can first select 2 students (in ${}^9{C}_2$ ways) and determine the ways in which the 2 medals can be distributed among them (in $2!$ ways).

This is given by

${}^9{C}_2\times 2!=72$

(iv)

One prize for Maths can be distributed in ${}^9{C}_1\times 1!$ ways,

two prizes for Physics can be distributed in $9C_2\times 2!$ ways and

three prizes for Chemistry can be distributed in ${}^9{C}_3\times 3!$ ways

such that no student gets more than one prize in the same subject.

Total number of ways

$={}^9{C}_1\times 1!$ ${\times }^9{C}_2\times 2!$ ${\times }^9{C}_3\times 3!$

$=9^3\times 8^2\times 7\times 2$

$=653184$