(a) (i) To form a four-digit number using the digits 1, 3, 5, and 6 with no repetition, we can arrange these 4 digits in 4! ways. Thus, the number of different four-digit numbers is \(4! = 24\).
(a) (ii) For odd numbers greater than 500, consider both 3-digit and 4-digit numbers:
3-digit numbers: The hundreds digit can be 5 or 6 (since the number must be greater than 500). The units digit must be odd (1, 3, or 5). Calculate the possibilities:
- 3-digit odd 500+: 4 ways
- 3-digit odd 600+: \(3 \times 2 = 6\) ways
4-digit numbers: The thousands digit can be 1, 3, 5, or 6. The units digit must be odd (1, 3, or 5). Calculate the possibilities:
- 4-digit odd 1000+: 4 ways
- 4-digit odd 3000+: 4 ways
- 4-digit odd 5000+: 4 ways
- 4-digit odd 6000+: 6 ways
Alternatively, consider the arrangement of digits for 4-digit odd numbers:
- Last digit in 3 ways, second to last in 3 ways, second in 2 ways, first in 1 way: 18 ways
Total: 28 ways
(b) Total arrangements of 6 cards is \(6! = 720\). If 4 and 5 are together, treat them as a single unit, so we have 5 units to arrange: \(5! = 120\). Within the unit, 4 and 5 can be arranged in \(2! = 2\) ways. Thus, the number of arrangements where 4 and 5 are together is \(5! \times 2 = 240\).
The number of ways 4 and 5 are not next to each other is \(720 - 240 = 480\).
Probability that 4 and 5 are not next to each other is \(\frac{480}{720} = \frac{2}{3}\).
