Exercise 1.60 (Bootstrap method)

Given $n\ge 2$ numbers $\left(a_1,a_2,\dots ,a_n\right)$ with no repetitions, a bootstrap sample is a sequence $\left(x_1,x_2,\dots ,x_n\right)$ formed from the $a_j$ ’s by sampling with replacement with equal probabilities. Bootstrap samples arise in a widely used statistical method known as the bootstrap. For example, if $n=2$ and $\left(a_1,a_2\right)=(3,1)$, then the possible bootstrap samples are $(3,3),(3,1),(1,3)$, and $(1,1)$.
(a) How many possible bootstrap samples are there for $\left(a_1,\dots ,a_n\right)$?
(b) How many possible bootstrap samples are there for $\left(a_1,\dots ,a_n\right)$, if order does not matter (in the sense that it only matters how many times each $a_j$ was chosen, not the order in which they were chosen)?
(c) One random bootstrap sample is chosen (by sampling from $a_1,\dots ,a_n$ with replacement, as described above). Show that not all unordered bootstrap samples (in the sense of (b)) are equally likely. Find an unordered bootstrap sample $b_1$ that is as likely as possible, and an unordered bootstrap sample $b_2$ that is as unlikely as possible. Let $p_1$ be the probability of getting $b_1$ and $p_2$ be the probability of getting $b_2$ (so $p_i$ is the probability of getting the specific unordered bootstrap sample $b_i$ ). What is $p_1∕p_2$ ? What is the ratio of the probability of getting an unordered bootstrap sample whose probability is $p_1$ to the probability of getting an unordered sample whose probability is $p_2$ ?