Final answer:
To find the number of odd integers from 1000 through 9999 with distinct digits, consider the digits separately for each place value and then divide by 2 since half will be prime and half will be composite.
Step-by-step explanation:
To find the number of odd integers from 1000 through 9999 with distinct digits, we need to consider the digits from the thousands place to the units place separately. For the thousands place, we have 9 possibilities (excluding 0). For the hundreds, tens, and units places, we have 9, 8, and 5 possibilities respectively (excluding the digit chosen for the thousands place, and any previous used digits). Therefore, the total number of odd integers with distinct digits is 9*9*8*5 = 3240. However, we need to divide this by 2 since half of the integers will be prime and half will be composite. So the final answer is 3240/2 = 1620.