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CAT 2020 Slot 3 — QA Question 1

Numbers (P&C)Easy

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

Answer & solution

Answer: 252

Solution

Easy

Counting "at least one repeated digit" directly is messy. Use complementary counting: subtract the numbers with all distinct digits from the total count of 3-digit numbers.

1

Total 3-digit numbers. The set is {100,101,…,999}\{100,101,\dots,999\}. The hundreds digit has 99 choices (11–99), the other two have 1010 each (00–99).

Total=9×10×10⇒ Total=900\begin{aligned} &\text{Total}=9\times10\times10\\ &\Rightarrow\ \text{Total}=900 \end{aligned}
2

Numbers with all distinct digits. Hundreds digit: 99 ways (11–99). Tens digit: 99 ways (any digit except the hundreds digit, 00 now allowed). Units digit: 88 ways (any except the two already used).

Distinct=9×9×8⇒ Distinct=648\begin{aligned} &\text{Distinct}=9\times9\times8\\ &\Rightarrow\ \text{Distinct}=648 \end{aligned}
3

Subtract. "At least one repeated digit" is the complement of "all distinct".

Repeated=900−648(from steps 1 and 2)⇒ Repeated=252\begin{aligned} &\text{Repeated}=900-648 \quad\text{(from steps 1 and 2)}\\ &\Rightarrow\ \text{Repeated}=252 \end{aligned}
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CAT 2020 Slot 3 QA Q1: How many integers in the set {100, 101, 102, ..., 999} have at least one digit repeated? — Solution | TheCATExam