"At least one" problems are everywhere: committees with at least one woman, passwords with at least one digit. Counting them head-on means many cases. Counting the opposite is usually one short step, built from the sum rule and combinations.
Key idea
Listen for the words "at least one," "some," "not all" and "contains a." Each is the opposite of a simple condition (none, all, never), and that opposite is usually a single product. A password with no digit is just a string of letters; a committee with no women is just a committee of men; a sequence of dice rolls with no 6 has $5$ choices per roll.
Why it works
Worked examples
Worked example: Committees with at least one woman
- All committees: $\binom{7}{3} = 35$.
- No women: all three from the $3$ men, $\binom{3}{3} = 1$.
- At least one woman: $35 - 1 = 34$.
Answer: $34$
Worked example: Bit strings
- All strings: $2^{8} = 256$. The only string with no 1 is 00000000, so at least one 1: $255$.
- "At least two 1s" is the opposite of "zero or one 1." There is $1$ string with no 1 and $8$ with exactly one (choose its position).
- At least two: $256 - 1 - 8 = 247$.
Answer: $255$ and $247$
Worked example: Repeated digits
- All PINs: $10^{4} = 10{,}000$.
- No repeated digit: $10 \cdot 9 \cdot 8 \cdot 7 = 5{,}040$.
- At least one repeat: $10{,}000 - 5{,}040 = 4{,}960$.
Answer: $4{,}960$
Common mistake