Q: How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two?

A: 112 numbers

There are four numbers that end with zero that satisfy this condition: 120 240 360 and 480. Each of these numbers can only be rearranged four ways (as a leading zero is not allowed) making 16 combinations.

Each of the numbers below can be arranged into 6 distinct numbers. For example, 213 can be 123 132 213 231 312 321. Below are 16 numbers and each can be rearranged in 6 ways. Therefore there are 96 three-digit numbers that satisfy this condition that do not have a zero in it.

The total is 96 + 16 = 112.

213

315

324

417

426

435

519

528

537

546

639

648

657

759

768

897