CAT 2020 Slot 3 QA Question 21: How many integers in the set $\{100, 101, 102, \ldots, 999\}$ have at least one digit repeated?

Question

How many integers in the set $\{100, 101, 102, \ldots, 999\}$ have at least one digit repeated?

Accepted Answer

Count the complement: three-digit numbers with no repeated digit.

Total three-digit numbers = $999 - 100 + 1 = 900$
Hundreds digit: 9 choices
Tens digit: 9 choices
Units digit: 8 choices
Numbers with no repeated digit = $9 	imes 9 	imes 8 = 648$

Numbers with at least one repeated digit = $900 -

How many integers in the set {100,101,102,…,999}\{100, 101, 102, \ldots, 999\}{100,101,102,…,999} have at least one digit repeated?

How many integers in the set $\{100, 101, 102, \ldots, 999\}$ have at least one digit repeated?