Level 4: Arranging, permutations

Counting and combinations · level 4 of 8 · rough age 11-12 (a guide for parents; anyone can start anywhere)

In level 3 you counted carefully, slot by slot. In this level those counts get names and formulas: the factorial \(n!\) for lining up everything, \({}^{n}P_{r}\) for lining up some, and a division whenever several orders look the same, because some things are identical or because a circle can be turned. You will learn the tools for arrangements with rules (glue, complement, fussy places first), and finish with the pigeonhole principle, a simple idea that proves surprising facts.

Lesson	You will be able to
Factorials	Use \(n!\) to count the ways to line up \(n\) things, explain why \(0! = 1\), and see how fast factorials grow
Arranging some	Count ordered choices like medals with \({}^{n}P_{r} = \dfrac{n!}{(n - r)!}\), and explain the division
Arranging with identical things	Count arrangements of BOOK, MISSISSIPPI and coin tosses by dividing by the swaps that change nothing
Arranging in a circle	Count seatings round a table, \((n - 1)!\), and necklaces that can be flipped, \(\dfrac{(n - 1)!}{2}\)
Arrangements with conditions	Handle "together", "not together", "in these places" and "ahead of" with glue, the complement and the gap method
The pigeonhole principle	Prove that a repeat must happen (birth months, socks, dots in a square) by choosing the right boxes

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