Find the sum of the four-digit greatest number and the five-digit smallest number, each number having three different digits.

For the greatest 4-digit number with three different digits:

Use 9 as many times as possible. Use 9 twice, then the next two largest different digits are 8 and 7.

Greatest 4-digit number = 9,987.

For the smallest 5-digit number with three different digits:

The ten thousand's place cannot be 0. Place 1 (smallest non-zero) at the ten thousand's place. Then 0s in remaining places. To have exactly three different digits, use one more digit (the smallest available, which is 2).

Smallest 5-digit number = 10,002.

Sum = 9,987 + 10,002

= 19,989

Hence, the required sum is 19,989.