Counting
How do you count the ways something can happen when there are far too many to list?
Where you are 10 inside
What this area is about
Listing every outcome works for a handful, not for the millions of codes a lock can take. This area counts without listing, by multiplying the choices made at each step and then correcting for what that multiplication counts twice or wrongly. The counts feed probability: when outcomes are equally likely, a chance is a count over a count.
How these fit together
The basic move. The fundamental counting principle multiplies the number of choices at each stage.
Ordered arrangements. Permutations of a whole set (n!), permutations of some objects (P(n, r)), and arrangements with a rule attached.
Choices where order does not matter. Combinations \binom{n}{r}; Pascal’s triangle, which builds them by addition; the binomial theorem, where they reappear in (a + b)^n; and combinations with repeats allowed.
Applications. Hypergeometric counting finds chances for draws without replacement, and the inclusion-exclusion principle counts the union of many overlapping sets.
A path through it
Counting in stages first, then permutations of a whole set, of some objects, and with restrictions. Combinations come next, then Pascal’s triangle and the binomial theorem, then combinations with repetition. Hypergeometric counting needs combinations and drawing without replacement. Inclusion-exclusion goes last: its proof uses the binomial theorem.
Finish here and the binomial distribution can count successes in repeated trials.