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Three-stage counting principle worksheets

Three mains, two sides and four drinks make twenty-four meals. These free printable worksheets extend the counting principle to three stages, where the outcomes are the product of all three.

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

The answer key prints on a separate page.

What's on this sheet

Three stages are the two-stage rule carried one step further: 3 × 2 × 4 = 24, every main with every side with every drink. Doing it two at a time keeps the picture — work out the 6 main-and-side combinations first, then give each of those a drink — so the “each one with each one” stays visible instead of collapsing into one long multiplication a child performs without seeing.

Why this sheet works

The order of the stages does not change the total, which is worth testing: 4 × 3 × 2 and 2 × 3 × 4 both give 24. Every answer is a whole count, so these sheets mark online as well as on paper, and the product is again the size of the sample space that a later probability will sit over.

How this one works

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

4 mains, 8 sides and 7 drinks. How many different meals can you make?

487
4×8=32
32×7=224
  1. Each of the 4 mains can go with every one of the 8 sides, so the choices multiply — they do not add.
  2. First two stages: 4 × 8 = 32.
  3. Then the 7 drinks: 32 × 7 = 224.

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