Problem
COM-B1-M01-P022 Subsets with Two Properties
#22
★★★★☆ Level 4 of 5
How many subsets of \(\{1,2,\ldots,20\}\) contain at least one even number and at least one multiple of \(5\)?
Subtract subsets with no even number and no multiple of \(5\), then add back the intersection.
There are \(2^{20}\) subsets. With no even number: choose only from \(10\) odd numbers, \(2^{10}\). With no multiple of \(5\): \(16\) available numbers, \(2^{16}\). With neither even numbers nor multiples of \(5\): \(8\) odd nonmultiples of \(5\), so \(2^8\). Answer: \(2^{20}-2^{10}-2^{16}+2^8\).
Serious complement example with intersecting forbidden conditions.