(a) Even numbers:
The units place can be filled by 0,2 or 8 this can be done 3 ways
∴ The number of ways is 5 × 6 × 6 × 3 = 540 ways.
(b) Number divisible by 5:
The units place can be filled by 0 or 5 in 2 ways
∴ the number of ways = 5 × 6 × 6 × 2 = 360 ways.
(c) greater than 5300:
Case 1: 5 is placed in thousands place. The 2nd place is done by (3 or 5 or 7 or 8) 4 ways
∴ the number of ways = 1 × 4 × 6 × 6 = 144 ways.
Case 2: If 7 or 8 is placed in thousandth place this can be done in 2 ways hundrenth place can be filled by any of the remaining six numbers
∴ The number of ways = 2 × 6 × 6 × 6 = 432 ways.
Hence the total number of ways is 432 + 144 = 576 ways.
(d) End with 7: Units place can be filled only one way the 1st place can be filled by 5 ways
∴ the number of ways = 1 × 5 × 6 × 6 = 180 ways.
