How many 4 digit positive integers have exactly 2 odd digits and 2 even digits?

The answer is 3375.

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If the first three digits are all odd, or all even, the number cannot contain $2$ even and $2$ odd digits. To find such numbers:

For each thousands digit, there are $5$ hundreds digits which match its parity, and for each of these, there are $5$ tens digits which also match parity. As there are $9$ possible thousands digits, $9 \times 5 \times 5 = 2250$ .

Excluding these leaves us with the numbers with the first three digits consisting of $2$ of one parity and $1$ of the other. Of these numbers, exactly half contain $2$ digits of each parity. (consider: each set of $10$ integers with a specific first $3$ digits contains $5$ possible final digits with the correct parity.)

And so, as there are $9000$ positive four-digit integers, $\frac{9000-2250}{2} = \boxed{3375}$ .