Answer :
Final answer:
There can be 20 even two-digit numbers constructed out of the digits 3, 4, 5, 6, 7 if digits can be reused, and 16 such numbers if digits cannot be reused.
Explanation:
when creating two-digit numbers, the number of possibilities can be determined by counting the available choices for each position. Here, since the two-digit number must be even, the last digit can either be 4 or 6. First, if digits can be reused, there are 5 choices for the first digit (3,4,5,6,7), and 2 choices for the final digit (4,6), so 5*2 = 10 two-digit even numbers. However, each number can be written in two ways (for example, 34 and 43), so we multiply by 2, getting 20 even two-digit numbers.
In the case where digits cannot be reused, there's still 2 choices for the final digit (4,6). However, the options for the first digit are now limited to 4 choices (since we've already used a digit), making 4*2 = 8. Multiplying by 2, since each number can be written in two ways, we get 16 even two-digit numbers.
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