IMC 2010 Q24
24. A new taxi firm needs a memorable phone number. They want a number which has a maximum of two different digits. Their phone number must start with the digit 3 and be six digits long. How many such numbers are possible?
- A 288
- B 280
- C 279
- D 226
- E 225
Intermediate Mathematical Challenge 2010, Question 24. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 280
There is the possibility of using only 3s giving one possible number 333333. Let's suppose a second digit is used, say \(x\). After the initial digit 3, there are 5 positions into which we can put either 3 or \(x\). So there are 2 choices in each of these 5 positions and so 25 = 32 possible choices – except that one such choice would be five 3s. So we get 31 choices. There are 9 possible values for \(x\), namely 0, 1, 2, 4, 5, 6, 7, 8, 9. So this gives 9 × 31 = 279 numbers. Together with 333333, this gives 280 numbers.
Solution © UK Mathematics Trust, from the IMC 2010 solutions.