Counting Principles: Systematic Listing

Methods in this video

Tap a time to jump to where the video teaches that method.

1. 1

   List in order: group first, then list each group

   ▶ 1:22

   When to use it: The question asks "how many ways / chains / different...". There aren't many possibilities, but there is no ready-made rule, so you have to find them one by one.

   • Writing whatever comes to mind makes you repeat some and miss others.
   • Group first (by number of steps, coins or toppings), list each group, then add the groups.
   • In each group, fix the first choice and go from smallest to largest, skipping none.
2. 2

   Count different results: the same result counts once

   ▶ 4:19

   When to use it: The question asks how many different amounts, totals or results you can make, and different choices can give the same result (identical coins or numbers, or small ones that add up to a bigger one).

   • Before you start, check what "different" means here. Are you counting ways, or results?
   • Write the result beside every way. Identical results count only once.
   • Swapping two identical things gives the same one.