SOLUTION: How many distinct even numbers less than 1000 can be formed by using only the numbers {0, 2, 5, 6, 9}; when repetition of a number is not allowed?

One digit numbers:

The even one-digit numbers are 0, 2, and 6.  That's 3 one-digit numbers. 

Two-digit numbers:

A. If we choose 0 for the second digit, then there are 4 choices for the first
digit.  That's 4 two-digit even numbers, 20, 50, 60, and 90. 

B. If we do not choose 0 for the second digit, then there are 2 choices for the
2nd digit (2 or 6).  That leaves 3 choices for the 1st digit.  That's 2×3 or 6
more even two-digit numbers, 52,92,56,96,26,56

Total number of two-digit numbers = 4+6 = 10

Three-digit numbers:

We can make each of the 10 two-digit numbers into a three digit number in 3
ways, by inserting any one of the remaining 3 digits between them.  So the
total number of three-digit even numbers is 10×3 = 30

Total: 3+10+30 = 43 even numbers, with no repeated digits, using only the
digits {0, 2, 5, 6, 9}.

Here are all 43:

0, 2, 6,
20, 26, 50, 52, 56, 60, 62, 90, 92, 96,
206, 250, 256, 260, 290, 296, 502, 506, 520, 526
560, 562, 590, 592, 596, 602, 620, 650, 652, 690
692, 902, 906, 920, 926, 950, 952, 956, 960, 962

Edwin