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

Solution :
The number of times the teacher goes to the zoo $$ = {\,^n}{C_3}.$$
The number of times a particular child goes to the zoo $$ = {\,^{n - 1}}{C_2}.$$
From the question, $$^n{C_3} - {\,^{n - 1}}{C_2} = 84$$
or, $$\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right) = 6 \times 84 = 9 \times 8 \times 7$$
$$ \Rightarrow \,\,n - 1 = 9.$$