Question

A father with $8$ children takes $3$ at a time to the Zoological gardens, as often as he can without taking the same $3$ children together more than once.

The number of times he will go to the garden is?

• A. $56$
• B. $21$
• C. $112$
• D. None of these

Answer 1

The correct option is A

$56$

The explanation for the correct option :

Step 1: Given conditions:

Number of children $\text{'}\mathrm{n}\text{'}=8$

Number of children taken at a time $\text{'}\mathrm{r}\text{'}=3$

Condition: Father visits the garden without taking the same $3$ children together more than once

Step 2: Calculate the number of possibilities:

Number of times he goes to the garden$={}_{}{}^{n}C_r$

Substitute the value of $\text{'}n\text{'}$and $\text{'}r\text{'}$in the formula ${}^{\mathbf{n}}\mathit{C}_{\mathbf{r}}\mathbf{}\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{r}\mathbf{)}\mathbf{!}\mathbf{}\mathbf{r}\mathbf{!}}$

We get ${}^{8}C_3=\frac{8!}{(8-3)!3!}$

$$\begin{gathered} =\frac{8!}{5!3!} \\ =\mathbf{56} \end{gathered}$$

Hence, Option(A) is the correct answer.