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Question 81

If the digits are not allowed to repeat in any number formed by using the digits 0, 2, 4, 6, 8, then the number of all numbers greater than 10,000 is equal to ___.


Correct Answer: 96

We must form numbers greater than 10,000 using the digits $$\{0, 2, 4, 6, 8\}$$ without repeating any digit. Since we have exactly 5 digits and the smallest 5-digit number formed from these digits (with a non-zero leading digit) is 20,468 which exceeds 10,000, every valid 5-digit number qualifies.

A 5-digit number uses all 5 available digits exactly once. The first digit cannot be 0 (otherwise it would not be a 5-digit number), so the first digit can be any of $$\{2, 4, 6, 8\}$$, giving 4 choices. The remaining 4 digits fill the other 4 positions in $$4! = 24$$ ways.

The total count is $$4 \times 24 = 96$$.

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