Question: In a class, there are 10 boys and 8 girls. The teacher wants to select either a boy or a girl to rep...

Question

In a class, there are 10 boys and 8 girls. The teacher wants to select either a boy or a girl to represent the class in a function. The number of ways the teacher can make this selection.
A. 18
B. 80
C. $^{10}{P_8}$
D. $^{10}{C_8}$

Answer 1

Solution

We will first calculate the total number of students by adding the number of boys and the number of girls in the class and then select using permutation and combination, one student from them.

Complete step-by-step answer:
We are given that we have 10 boys and 8 girls in a class.
Therefore, total number of students = number of boys + number of girls
So, total number of students = 10 + 8 = 18
Now, we have to select one student among the 18 students of the class.
Now, we can select any one among those 18 students.
So, the number of choices we have is $^{18}{C_1}$ .
Now, we will use the formula given by the following expression:-
${ \Rightarrow ^n}{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$
Replacing n by 18 and r by 1, we get:-
${ \Rightarrow ^{18}}{C_1} = \dfrac{{18!}}{{1!(18 - 1)!}}$
Simplifying the calculations a bit, we will get:-
${ \Rightarrow ^{18}}{C_1} = \dfrac{{18!}}{{17!}}$
Now, since we know that n! = n.(n-1).(n-2)…….1
So, we get:-
${ \Rightarrow ^{18}}{C_1} = \dfrac{{18 \times 17!}}{{17!}}$
Simplifying the calculations further to get the result:-
${ \Rightarrow ^{18}}{C_1} = 18$

Hence, the correct answer is (A) 18.

Note:
The students must note that the given information about boys and girls separately is there to confuse us so that, we get tangled in between that but we just need to select a student (be it a boy or a girl), so, we first calculated the sum of number of students collectively and then choose one from them.
The students must also note that, here we did not even need to use the combinations as we did in the above solution. For example:- If you have 10 different types of chocolates and your mother is allowing you to eat only one of them, then you can select any one of those 10 chocolates in 10 ways.
Similar thing happened in the above solution. We had 18 students to select one from.