Counting principles

Kofi has $4$ shirts, $3$ pairs of trousers and $2$ pairs of shoes. An outfit is one shirt, one pair of trousers and one pair of shoes. How many different outfits can he make?

A school library in Quito has $12$ novels and $8$ comics on a display shelf. Ana may borrow one book from the shelf. In how many ways can she choose?

In how many different orders can $6$ different books be arranged on a shelf?

Ten swimmers race in a final. In how many ways can the gold, silver and bronze medals be awarded? (No ties.)

How many 4-digit PINs (from $0000$ to $9999$) contain at least one repeated digit?

How many of the whole numbers from $1$ to $100$ are divisible by $3$ or by $5$ (or both)?

Which of these are good reasons for defining $0! = 1$? (Choose all that apply.)

A code has $3$ symbols, each a capital letter ($26$ choices) or a digit ($10$ choices). How many codes contain at least one letter and at least one digit?

In the Lean file, perm n k is $P(n, k)$ and fact n is $n!$. Which statement correctly formalises $P(n, k) = \tfrac{n!}{(n - k)!}$?

Write a recursive function arrangements(items) that returns a list of all orderings of the list items (each ordering a list), without using itertools. Idea: for each item, put it first and follow it by every arrangement of the remaining items. Then check that $4$ items give $4! = 24$ different arrangements.