Section 3.2

Dice, Coins, Urns and Seats

Counting favourable outcomes for dice, coins, urns, random lineups and seatings (adjacent birds, age order around a table, runs of one kind).

Toolkit

Equally likely outcomes reduce to counting favourable configurations among a fixed sample space; the key is choosing the right combinatorial model (ordered vs. unordered, with vs. without replacement, labelled vs. unlabelled positions) and then counting cleanly via symmetry, cases, bijections, or element-by-element independence.

Useful tools:

Positions/adjacency: count adjacent pairs or runs directly, often by “gap” or “anchor” arguments; circular arrangements fix one object to kill rotational symmetry, giving $(n-1)!$ total orders.
Element-wise independence: when a global event decomposes into independent per-element conditions, multiply per-element probabilities.
Casework on a pivotal quantity (first outcome, extreme value, parity, which symbol repeats) collapses large counts into short enumerations.
Symmetric dice/coin sums: exploit symmetry $k \leftrightarrow n+1-k$ or $p_i \leftrightarrow p_{n+1-i}$ .
Multinomial and binomial coefficients count multiset arrangements and letter/pattern frequencies: $\binom{n}{k_1,\dots,k_m}/m^n$ .
Convert combinatorial conditions on integers into bounded inequalities (e.g. $(x-1)(y-1)\le 1$ ) to shrink the search.

Problems