How many 4-digit numbers divisible by 5 can be formed using the digits 0, 1, 2, 3, 5 and 7, if no digit is repeated in a number?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 108
A number is divisible by 5 when its units digit is 0 or 5. A 4-digit number cannot begin with 0, so the two cases are counted separately and added.
Case 1: units digit 0
The other three places take digits from 1, 2, 3, 5, 7 (0 is used up)
Thousands, hundreds, tens: \(5\times4\times3=60\)
Case 2: units digit 5
Digits left: 0, 1, 2, 3, 7
Thousands place: not 0, so 4 choices (1, 2, 3 or 7)
Hundreds place: 4 choices left (0 is now allowed)
Tens place: 3 choices left
Numbers \(=4\times4\times3=48\)
Total \(=60+48=108\)
Why the other options are incorrect
Option (1) 120 — Incorrect. This is \(2\times60\); it counts the units digit 5 case like the 0 case and so lets numbers begin with 0.
Option (3) 60 — Incorrect. This counts only the numbers ending in 0 and leaves out those ending in 5.
Option (4) 96 — Incorrect. This is \(2\times48\); the numbers ending in 0 have no first-digit problem and number 60, not 48.