Question

In a class there are 27 boys and 14 girls. The teacher 2 wants to select 1 boy and 1 girl to represent the class in a function. In how many ways can the teacher make this selection?

Answer 1

Hint: In order to solve the given problem use the concept of permutation and combination and further combine both the events as they are exhaustive events. Solve the combination
by the use of formula to get the final answer.

Complete step by step answer:
Given that:
Number of boys in the class is 27.
Number of girls in the class is 14.
The teacher needs to select a boy and a girl for the function. To find the number of ways this can be done.
As we know from the concept of combination that the number of ways of selection of n items from m different items is ${}^m{C_n}$ .
Let us use the above formula further.
The number of ways of selecting a boy from 27 boys in the class is:
${}^{27}{C_1}$
And the number of ways of selecting a girl from 14 girls in class is:
${}^{14}{C_1}$
As we know that both the event is exhaustive so to find the no of ways we have to multiply the individual no of ways.
So, number of ways of selecting the boy and girl is:

\[   = {}^{27}{C_1} \times {}^{14}{C_1} \]

\[   = \dfrac{{27!}}{{26! \times 1!}} \times \dfrac{{14!}}{{13! \times 1!}} \]

\[   = 27 \times 14 \]

\[  = 378  \]

Hence, the number of ways in which teachers can make a selection is 378.

Note: In order to solve such types of problems students must use the methods of permutation and combination. The problem can also be solved by the method of factorial but that method will be more tiresome than this one. Students must remember that we use combinations in order to find out the number of selections out of a given number of items and we use permutation in order to find the total number of arrangements out of the given number of items.