Question

Of the three-digit integers greater than 700, how many have two digits that are equal to each other and the remaining digit different from the other two?

• A. 90
• B. 82
• C. 80
• D. 45
• E. 36

Hint:

For counting problems with multiple constraints, break the problem into smaller, non-overlapping cases. Handle each case carefully, paying close attention to boundary conditions like "greater than 700," which may affect one case differently than others.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We need to count the number of integers between 701 and 999 (inclusive) that have exactly two identical digits. We can break this problem down by considering the possible hundreds digit (7, 8, or 9) and the pattern of the digits (e.g., AAB, ABA, BAA).
Step 2: Detailed Explanation:
We will analyze the count for each hundred's range separately. The condition is that two digits are equal, and one is different. Case 1: Numbers in the 700s (from 701 to 799)
The first digit is fixed as 7.

Pattern 77X: The repeated digit is 7. The third digit, X, can be any digit from 0 to 9 except 7 (to avoid 777). So, X can be \{0, 1, 2, 3, 4, 5, 6, 8, 9\}. This gives 9 numbers.
Pattern 7X7: The repeated digit is 7. The middle digit, X, can be any digit except 7. So, X can be \{0, 1, 2, 3, 4, 5, 6, 8, 9\}. This gives 9 numbers.
Pattern 7XX: The repeated digit is X, which cannot be 7. X can also not be 0, as the number must be greater than 700 (700 is not included). So X can be \{1, 2, 3, 4, 5, 6, 8, 9\}. This gives 8 numbers.
Total for the 700s = \(9 + 9 + 8 = 26\).
Case 2: Numbers in the 800s (from 800 to 899)
The first digit is fixed as 8.

Pattern 88X: X can be any digit except 8. This gives 9 numbers.
Pattern 8X8: X can be any digit except 8. This gives 9 numbers.
Pattern 8XX: X can be any digit except 8. This gives 9 numbers (X can be 0 here).
Total for the 800s = \(9 + 9 + 9 = 27\).
Case 3: Numbers in the 900s (from 900 to 999)
The first digit is fixed as 9.

Pattern 99X: X can be any digit except 9. This gives 9 numbers.
Pattern 9X9: X can be any digit except 9. This gives 9 numbers.
Pattern 9XX: X can be any digit except 9. This gives 9 numbers.
Total for the 900s = \(9 + 9 + 9 = 27\).
Step 3: Final Answer:
The total number of such integers is the sum of the counts from all cases: \[ \text{Total} = 26 + 27 + 27 = 80 \]