Level 3: Counting with care

Counting and combinations · level 3 of 8 · rough age 9-10 (a guide for parents; anyone can start anywhere)

In level 2 you learned to count by multiplying along a tree. In this level you learn to count carefully: to notice when a choice cannot be repeated, to deal with the awkward part of a question first, and to count the things you do not want when that is easier. You will count every square on a chessboard, find out when turning a table makes two seatings "the same", and see that counting subsets and counting factors are the same idea.

Lesson	You will be able to
Repeats allowed or not	Count codes and words with and without repeats, and explain why each slot then has one choice fewer
Fussy parts first	Count numbers with rules (even, no 0 in front, different digits) by filling the fussiest slot first and splitting into cases
Counting what you do not want	Count "at least one" by taking the complement away from the total, and explain why simple adding over-counts
Squares on a chessboard	Count squares of every size (204 on a chessboard) and rectangles by choosing lines (1296)
Counting by symmetry	Count seatings at a round table and necklaces, and know when dividing by the number of turns works
Factors and subsets	Count subsets with "in or out" (\(2^n\)) and count the factors of a number from its primes

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