Digit arrangements
Digit questions are ordinary position-counting questions with extra rules. Decide first whether the object is a number or a code, whether a digit may be reused, and which position is most restricted.
Use the digits 0, 1, 2, 3 and 4.
Fix the last digit first, then protect the first position.
Last digit 0
4 × 3 × 2 = 24
The first digit can be 1, 2, 3 or 4.
Last digit 2 or 4
2 × 3 × 3 × 2 = 36
Zero is still blocked from the first position.
total = 24 + 36 = 60
Separate numbers, codes and repeated digits
- A multi-digit number cannot begin with 0. A code may begin with 0 unless the question forbids it.
- Without repetition means a chosen digit is removed, so the next choice count falls.
- Repetition allowed means a digit may be reused. If every one of symbols can fill each of positions, the count is .
Common mistake
Use the last digit to control odd, even and multiples
- An even number ends in 0, 2, 4, 6 or 8; an odd number ends in 1, 3, 5, 7 or 9.
- A multiple of 5 ends in 0 or 5; a multiple of 10 ends in 0.
- Fix the last digit first. If one last-digit choice changes the choices for the first digit, split into separate cases.
Key idea
Translate a bound into a first-digit condition
Worked example
Odd four-digit numbers greater than 3000
Use four of the digits 0, 1, 2, 3 and 4, without repetition.
- If the first digit is 3, the last digit must be 1. Arrange two of the remaining three digits in the middle: .
- If the first digit is 4, the last digit can be 1 or 3. For each choice, arrange two of the remaining three digits: .
How many even four-digit numbers can be made from the digits 0, 1, 2, 3, 4 and 5, with no digit repeated?
Show worked answer
Split into last digit 0, and last digit 2 or 4. In the second case, zero is still unavailable for the first position.