IMC 2014 Q24
24. How many 4-digit integers (from 1000 to 9999) have at least one digit repeated?
- A 62 × 72
- B 52 × 72
- C 52 × 82
- D 42 × 82
- E 42 × 92
Intermediate Mathematical Challenge 2014, Question 24. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A 62 × 72
There are 9000 four-digit integers. To calculate the number of these which have four different digits, we note that we have a choice of 9 for the thousands digit. We now have a choice of 9 for the hundreds digit (since we can choose 0 as a possible digit). After these two digits have been chosen, we have a choice of 8 for the tens digit and then 7 for the units digit. So the number of four-digit numbers in which all digits are different is 9 × 9 × 8 × 7.
Therefore the number of four-digit numbers which have at least one digit repeated is 9000 − 9 × 9 × 8 × 7 = 9(1000 − 9 × 8 × 7) =
9 × 8 × (125 − 9 × 7) = 72 × (125 − 63) = 72 × 62.
Solution © UK Mathematics Trust, from the IMC 2014 solutions. Longer versions with investigations: Solutions and investigations (PDF).