Worked solution
Allow leading zero in a six-position PIN
Joshua tries to recall the 6-digit PIN number for his ATM card. Assume 000000 is even and valid. How many possible numbers can there be if he remembers that:
(i) the number starts and ends with the digit 5? mark
(ii) the number is odd and the digits do not repeat? marks
(iii) the digits do not repeat and there are exactly 3 odd digits? marks
[2011/PJC/II/10]
CJC · Review problems · Question 3
Answer
(i) 10000; (ii) 75600; (iii) 72000.
(i) Fix the first and last digits
The four middle positions each have ten choices. Repetition is not excluded in this part, so .
(ii) Choose an odd last digit first
The final position has five choices: 1, 3, 5, 7 or 9. Fill the other five positions from the remaining nine digits without repetition. The source product, read left to right after filling the last position first, is . A leading zero is permitted by the PIN convention.
(iii) Select the six distinct digits, then arrange
Choose three odd digits from five and three even digits from five (including zero): for each selection. All six selected digits are distinct, so .
Common mistakes
Do not apply an ordinary integer’s leading-zero exclusion to this explicitly stated PIN convention.
Solution notes
All source working and printed marks retained. The explicit 000000 convention is retained in the prompt.