Let $U=1,2,3,4,5,6,7,8,9,10$. Define $A=1,3,5,7,9$ and $B=2,4,6,8,10$. How many subsets of $U$ contain at least one element from $A$ and at least one element from $B$?

Step-by-step solution

1. Total subsets of $U$: $2^{10}=1024$.
2. Subsets with no element of $A$: choose only from $B$ (5 elements), giving $2^5=32$ subsets.
3. Subsets with no element of $B$: choose only from $A$ (5 elements), also $2^5=32$ subsets.
4. Subsets with no elements from either $A$ or $B$: only the empty set – $1$ subset.
5. By inclusion–exclusion: $1024-32-32+1=961$ subsets satisfy the requirement.
6. Thus the answer is $961$.

Concept tested: inclusion–exclusion principle for subsets

Published 13 October 2026 · How this solution was prepared and checked