Question

In class, there are $15$ boys and $10$ girls. How many ways can a teacher select $1$ boy and $1$ girl to represent the class at a seminar?

Answer 1

Hint:Combinations is used if the certain objects are to be arranged in such a way that the order of objects is not important, here we will apply the combinations concepts since any one boy and one girl needs to represent the class irrespective of any order to be followed and then will find the resultant required value.

Complete step by step answer:
Here we will use the formula for the combinations in such a way that the order of the object is not important. The number of ways a teacher can select $1$ boy and $1$ girl to represent the class seminar can be given by using the combinations formula –
${}^{15}{C_1} \times {}^{10}{C_1}$
Simplify the above expression –
$ 15 \times 10$
Multiply the above terms –
$ 150$ ways

Hence, a teacher can select $1$ boy and $1$ girl to represent the class at a seminar in 150 ways.

Note:Always Know the difference between the permutations and combinations and apply its concepts and its formula accordingly. In permutations, the specific order and arrangement is the most important while in combination the certain objects are to be arranged in such a way that the order of objects is not important.
-Formula for combinations is given by$^nc{}_r = \dfrac{{n!}}{{r!(n - r)!}}$
-Formula for the permutations is given by ${}^np{}_r = \dfrac{{n!}}{{(n - r)!}}$