Question

In a city no person has 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. $2^{32}-1$
• C. $2^{32}+1$
• D. None of these

Answer 1

The correct option is B $2^{32}-1$

In the mouth, there are $32$ places for the teeth.

For each place, two choices are possible, i.e., either tooth or not.

Choice for $1$ teeth $=2$

Similarly choices for $2$ teeth $=2\times 2=2^2$

Choice for $3$ teeth $=2\times 2\times 2=2^3$

.....

.....

Hence, choice for $32$ teeth $=2^{32}$

But this includes one case where there is no teeth.

$\therefore$ Required number of ways $2^{32}-1$