Section 1.1

Arrangements and Selections

Direct counts with the product rule, permutations and combinations, including restrictions (adjacency, fixed positions, monotone patterns), repeated letters (multinomial coefficients) and assignments of people to teams or seats.

Toolkit

Most problems reduce to the product rule plus two counting primitives: permutations $P(n,k)=n!/(n-k)!$ and combinations $\binom{n}{k}=n!/(k!(n-k)!)$ , together with the multinomial coefficient for arranging a multiset with repeated types:

\binom{n}{n_1,\dots,n_r}=\frac{n!}{n_1!\cdots n_r!}.

Key techniques:

Gap/slot method: arrange unrestricted items, then insert restricted ones into the gaps created, or choose which of $n$ positions a subset occupies when only relative order matters ( $\binom{n}{k}$ ways).
Fix-then-free: satisfy a constraint on a special element or subgroup first, then count the remaining unconstrained arrangement independently.
Symmetry: if all positions are equally likely for a labelled item among $n$ by relabelling symmetry, its count is the total divided by $n$ .
Palindromes/unimodal shapes: the structure is determined by one half (or one side) of the choice, reducing to arranging a smaller multiset.
Casework: split on a parameter (shared letter, group size, parity) and sum binomial/multinomial counts over cases.

Problems