Question

The kindergarten teacher has 30 kids in her class. She takes 6 of them at a time, to a zoological garden as often as she can, without taking the same 6 kids more than once. Then the number of visits, the teacher makes to the garden exceeds that of a kid by

Answer 1

$\displaystyle \binom{30}{6} - \binom{29}{5}$

Answer 2

For 30 kids taken 6 at a time:

• Total visits by the teacher = $\binom{30}{6}$.
• For a particular kid, the number of visits in which they are included = $\binom{29}{5}$ (since we choose the remaining 5 kids from the other 29).

Thus, the teacher’s visits exceed that of a kid by:

$$\binom{30}{6} - \binom{29}{5}.$$