How many integers between 100 and 300 begin with 2 or end with 2?
2011
How many integers between 100 and 300 begin with 2 or end with 2?
Answer: B. 110 — ConceptFor two sets A and B, the inclusion–exclusion principle gives |A ∪ B| = |A| + |B| − |A ∩ B|. The overlap must be subtracted once because it is included…
- A.
100
- B.
110
- C.
120
- D.
180
Show answer & explanation
Correct answer: B
Concept
For two sets A and B, the inclusion–exclusion principle gives |A ∪ B| = |A| + |B| − |A ∩ B|.
The overlap must be subtracted once because it is included in both individual counts.
Application
Let A be the integers from 100 to 300 that begin with 2. These are 200 through 299, so |A| = 100.
Let B be the integers from 100 to 300 that end with 2. These are 102, 112, …, 292, so |B| = 20.
The overlap A ∩ B consists of 202, 212, …, 292, so |A ∩ B| = 10.
Therefore, |A ∪ B| = 100 + 20 − 10 = 110.
Cross-check
The 100 integers from 200 through 299 already include the 10 integers in that block that end with 2.
Outside that block, exactly 10 integers end with 2: 102 through 192. Thus 100 + 10 = 110.
Hence, the required number of integers is 110.