Since order matters, we should not use the word "combination", In
algebra, the word "combination" is used only when the order does
not matter.
But the order does matter here, so we should use the word "permutation"
instead of "combination"
Case 1: None of the digits are the same:
These are the arrangements of {0,2,4}
Choose the first digit 3 ways
Choose the second digit 2 ways
Choose the third digit 1 way.
3×2×1 = 6 ways for case 1. Those 6 are
024, 042, 204, 240, 402, 420
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Case 2: Exactly two 0's
Choose the remaining digit 2 ways. {2,4}
We can arrange them in 3!/2! = 3 ways
That's 2×3 = 6 for case 2. Those 6 are:
002, 004, 020, 040, 200, 400
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Case 3. Exactly 2 2's:
Choose the remaining digit 2 ways. {0,4}
We can arrange them in 3!/2! = 3 ways
That's 2×3 = 6 for case 3. Those 6 are:
022, 224, 202, 242, 220, 422
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Case 4. All 3 2's:
That's just 1 way, 222.
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Grand total = 6+6+6+1 = 19 ways.
Edwin
