The smallest 4-digit number having three different digits is
1102
1012
1020
1002
Goal: Make the smallest 4-digit number that uses exactly three different digits (one digit will repeat).
Step 1: For the smallest 4-digit number, the thousands place must be \(1\) (it cannot be \(0\)).
So we start with: \(1\_\_\_\).
Step 2: To keep the number as small as possible, put \(0\) in the hundreds place:
\(10\_\_\).
Step 3: Still keep it small: put \(0\) in the tens place:
\(100\_\).
Step 4: Right now, the digits used are \(\{1, 0\}\) → only two different digits.
We need three different digits, so the units digit must be the smallest digit not used yet and different from \(1\) and \(0\). That is \(2\).
Step 5: Put \(2\) in the units place:
\(1002\).
Check: Digits in \(1002\) are \(1, 0, 0, 2\). Distinct set is \(\{1, 0, 2\}\) → exactly three different digits, and the number is as small as possible.
Answer: \(1002\) (Option D).