Math PicnicFree printable practice

Two-stage counting principle worksheets

Four shirts and three pairs of shorts make twelve outfits, not seven. These free printable worksheets count the outcomes of a choice made in two stages — the multiplication rule behind every sample space.

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Two-stage counting principle
Two-stage counting principle worksheet — Two independent stages, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

The one idea is that independent choices MULTIPLY. Each of the four shirts can pair with each of the three shorts, so the outfits are 4 × 3, and adding to get 7 is the mistake the sheet is built around. A child who pictures every shirt matched with every pair of shorts — a grid of pairings — sees straight away why it is a product and not a sum.

Why this sheet works

Every answer is a whole number, so unlike the probabilities this counting later supports, these sheets can be marked on paper and on screen alike. The product is exactly the size of the sample space: knowing there are 12 equally likely outfits is what a probability about outfits will later be written over.

How this one works

One question from this sheet, worked through a step at a time.

4 mains and 6 sides. How many different meals can you make?

46
4×6=24
  1. Each of the 4 mains can go with every one of the 6 sides, so the choices multiply — they do not add.
  2. 4 × 6 = 24.

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