How many different positive integers can be made from the digits { 2,4,6,8} if repetitions are allowed?

I agree with Myles. The question is flawed. However, the possibilities are finite.

1 digit positive integers = 4¹ = 4 (i.e. 2, 4, 6, 8)
2 digit positive integers = 4² = 16 (i.e. 22, 24, 26, 28, 42, 44, 46, 48, 62, 64, 66, 68, 82, 84, 86, 88)
3 digit positive integers = 4³ = 64
4 digit positive integers = 4⁴ = 256

Total positive integers = 4 + 16 + 64 + 256 = 340

However, it seems the question implies 4 digit integers only. Then answer is 256

The question is flawed. They are expecting the answer 256, which is the number of positive integers with 4 digits. Each digit can be one of the 4 options, so 4 x 4 x 4 x 4 = 256. However, the question just asks for positive integers. There are also single-digit integers 2, 4, 6, and 8. Two-digit integers (16 of them) . . . and so on. So, the answer to the question asked is infinite.

There is a countable infinity of these such numbers. Just like there are a countable infinity of numbers with 7's in them.