91Ó°ÊÓ

A carousel has eight seats, each representing a different animal. Eight girls are seated on the carousel facing forward (each girl looks at another girl's back). In how many ways can the girls change seats so that each has a different girl in front of her? How does the problem change if all the seats are identical?

Let \(n\) be a positive integer and let \(p_{1}, p_{2}, \ldots, p_{k}\) be all the different prime numbers that divide \(n\). Consider the Euler function \(\phi\) defined by$$ \phi(n)=|\\{k: 1 \leq k \leq n, \mathrm{GCD}\\{k, n\\}=1\\}| .$$ Use the inclusion-exclusion principle to show that $$\phi(n)=n \prod_{i=1}^{k}\left(1-\frac{1}{p_{i}}\right) .$$

Prove that the convolution product satisfies the associative law: \(f *(g * h)=\) \((f * g) * h .\)