Mathematics from zero

Combinatorics: free-recall review

Crux Free-recall prompts across the combinatorics unit. Answer each in your own words first, then reveal the model answer and compare.

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◷ 14 min

Retrieval beats re-reading. For each prompt, say or write a full answer from memory before you open the model answer — the effort of recall is what makes the rules stick, not seeing them again.

Goal

Reconstruct the unit’s core ideas — why counting means multiplying, what a factorial counts, how permutations and combinations differ, and when to divide by a factorial — without looking back at the lessons.

Recall before you leave

01
State the counting principle, and explain why the option counts multiply rather than add.
02
What is a factorial, and what does n! count?
03
Define a permutation, and explain why the choices shrink by one at each position.
04
Define a combination, and explain how it relates to a permutation.
05
You face a counting problem. What is the first question to ask, and how does the answer route you to the right tool?
06
Why does choosing all n items as an unordered group give exactly 1, even though arranging all n gives n!?

Recap

If you could reconstruct each answer from memory, you hold the unit’s spine: the counting principle multiplies independent choices because each early option opens a fresh set of later ones; a factorial counts the orderings of a whole set; a permutation multiplies shrinking factors because placed items are used up; and a combination divides the permutation count by r! to merge the orderings order does not care about. Every problem starts with two questions — separate choices or one set, and does order matter.

Continue the climb ↑Combinatorics: build a counting handbook