Question

In a city no two 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 the maximum population of the city is

• A. $2^{32}$
• B. ${\left(32\right)}^2–1$
• C. $2^{32}–1$
• D. $2^{32-1}$

Answer 1

The correct option is C

$2^{32}–1$

Explanation for correct option:

Find the maximum population.

There are total of $16$pairs of teeth, i.e. equal to $32$ teeth.

For each place we have two choices, either there is a tooth or there is no tooth thus, number of ways of filling $32$ places $=2^{32}$

Since there is no person without a tooth, the maximum population becomes,
$$\begin{gathered} =2^{32}-2^0 \\ =2^{32}-1 \end{gathered}$$

The correct option is C.