Question

In a city no persons have identical set of teeth and there is no person without a tooth. Also no person has more than $32$ teeth. If we disregard the shape and size of tooth and consider only the positioning of the teeth, then let $k$ be the maximum population of the city. If $k=n^m-1$. Find $\frac{m}{n}$.

Answer 1

There are $32$ places for the teeth in the mouth. For each place, we have two choices either there is a tooth or there is no tooth.
Choices of $1$ teeth $=2$
Again Choices of $2$ teeth $=2\times 2=2^2$
Choices of $3$ teeth $=2\times 2\times 2=2^3$
Similarly Choices of $32$ teeth $=2^{32}$
which also includes person without teeth
But given that no person is without a tooth
Hence, the maximum population of the city $=2^{32}-1$.
Now $k=n^m-1$
So $m=32$ and $n=2$
Hence, $\frac{m}{n}=16$