IMC 2015 Q20
20. A voucher code is made up of four characters. The first is a letter: V, X or P. The second and third are different digits. The fourth is the units digit of the sum of the second and third digits.
How many different voucher codes like this are there?
- A 180
- B 243
- C 270
- D 300
- E 2700
Intermediate Mathematical Challenge 2015, Question 20. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 270
There are 3 different possibilities for the first character. The second character may be any digit from 0 to 9 inclusive, so it has 10 different possibilities. The third character differs from the second digit, so has 9 different possibilities. Once the second and third characters are determined, the fourth character is also determined since it is the units digit of the sum of the second and third characters.
So, the number of different codes is 3 × 10 × 9 = 270.
Solution © UK Mathematics Trust, from the IMC 2015 solutions. Longer versions with investigations: Solutions and investigations (PDF).