Count What You Don't Want
When the desired set is awkward but its opposite is simple, count the complete universe and remove the complement.
What you'll learn
- Recognize when direct groups overlap or become harder than the original count.
- Describe the complete opposite of a desired set and choose the correct whole universe.
- Count the whole, count the opposite, and subtract to find the desired number of outcomes.
Before you start
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Worked recap: Count three-digit numbers containing a zero
There are 9 × 10 × 10 = 900 three-digit numbers. The first digit cannot be zero.
There are 9 × 9 × 9 = 729 with no zero anywhere. Subtract this complement: 900 − 729 = 171 contain at least one zero.