Question

A teacher takes 3 children from her class to the zoo at a time as often as she can, but she does not take the same three children to the zoo more than once. She finds that she goes to the zoo 84 times more than a particular child goes to the zoo. The number of children in her class is

• A. 12
• B. 10
• C. 60
• D. none of these

Answer 1

Answer: (B)

10

Number of ways a teacher can go is the number of ways group of 3 can be made from$n$ students ${=}^n{C}_3$
Number of ways a particular child is chosen = number of ways 2 students can be selected from remaining$(n-1)$ students ${=}^{n-1}{C}_2$
Given,
${}^n{C}_3{-}^{n-1}{C}_2=84$
$\frac{n!}{3!(n-3)!}-\frac{(n-1)!}{2!.(n-3)!}=84$
$\frac{n(n-1)(n-2)}{6}-\frac{3(n-1)(n-2)}{6}=84$
$(n-1)(n-2)(n-3)=84.6=\mathrm{12.7.6}=\mathrm{4.3.7.3.2}=\mathrm{9.8.7}$
So, $n=10$
Hence, option 'B' is correct.