PK 2017 Q20
20. Frederik wants to make a special seven-digit password. Each digit of his password occurs exactly as many times as its digit value. The digits with equal values always occur consecutively, e.g. 4444333 or 1666666. How many possible passwords can he make?
- A 6
- B 7
- C 10
- D 12
- E 13
Pink Kangaroo 2017, Question 20. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 13
The password has length 7 so the different digits making it up must add to 7. The possibilities are: {7}, {6, 1}, {5, 2}, {4, 3}, {4, 2, 1}. Using only the digit 7 produces just one password, 7777777. Using two digits gives two possibilities, depending on which digit goes first, so the three pairs give 2 × 3 = 6 phone numbers. Three different digits can be arranged in six ways. This gives 1 + 6 + 6 = 13 possibilities.
Solution © UK Mathematics Trust, from the PK 2017 solutions.