Problem 11 · 2008 AMC 8
Easy
Counting & Probability
inclusion-exclusion
Each of the 39 students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and 26 students have a cat. How many students have both a dog and a cat?
Show answer
Answer: A — 7.
Show hints
Hint 1 of 2
Add 20 dog-owners + 26 cat-owners and you get 46 — but only 39 kids exist, so somebody got counted twice.
Still stuck? Show hint 2 →
Hint 2 of 2
Anyone counted twice owns both pets. The overcount itself is the answer.
Show solution
Approach: the overcount is the overlap
- Counting dog-owners (20) and cat-owners (26) separately gives 20 + 26 = 46 "pet slots," but there are only 39 kids. The extra 46 − 39 = 7 came from kids who got tallied in both groups.
- Those double-counted kids are exactly the ones with both a dog and a cat: 7.
- Why this transfers: this is inclusion–exclusion — |A| + |B| counts the overlap twice, so |both| = |A| + |B| − |either|. You'll reuse it for any "how many in both groups" question.
Mark:
· log in to save