Complementary counting

Complementary counting is counting the complement of the set we want to count, and subtracting that from the total number of possibilities, or the universal set for that particular problem. A big hint that complementary counting may lead to a quick solution is the phrase "at least".

Video

This is a video explaining the basics of casework, complementary counting, and overcounting (PIE): https://youtu.be/Zhsb5lv6jCI

Examples

Introductory

• 2006 AMC 10A Problem 21
• 2002 AIME I Problem 1
• 2008 AMC 12B Problem 22

Somewhat Harder

• 2004 AIME I Problems/Problem 15

See also

• Combinatorics
• Probability