A 5-digit number is made using the digits 0, 1, 4, 5, 6, 7 and 9. No digit may be used more than once in any 5-digit number. Find how many such 5-digit numbers are even and greater than 50000.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

1. [B1] Starting with a 6: 120 ways
2. [B1] Starting with 5, 7 or 9: 540 ways
3. [B1] Total 660
4. [(B1)] Or Alternative 1 Ending with a 6: 180 ways
5. [(B1)] Ending with 0 or 4: 480ways
6. [(B1)] Total 660
7. [(B1)] Or Alternative 2 11 ways of obtaining even 5-digit numbers which start with 5, 6, 7, 9
8. [(B1)] ⁵P₃ ways of arranging remaining 3 digits: 60
9. [(B1)] 11×60=660
10. [(B2)] Or Alternative 3 Total arrangements ⁷P₅ minus (all odds + evens starting with 1 + evens starting with 0 or 4) = 2520 – (1440 + 180 + 240)
11. [(B1)] 660
12. [B1] n!/((n-4)!4!) = 6n!/((n-2)!2!)
13. [B1] for (n-2)(n-3)
14. [B1] for 72
15. [M1] for correct attempt to form and solve a quadratic equation
16. [A1] for n = 11 only