How many distinct 8-digit numbers can be formed by rearranging the digits of the number 11223344 such that odd digits occupy odd positions and even digits occupy even positions?

To find the number of distinct 8-digit numbers, we must arrange the given digits into their assigned positions based on the parity rule.

The number 11223344 consists of four odd digits (1, 1, 3, 3) and four even digits (2, 2, 4, 4).

An 8-digit number has four odd positions (1st, 3rd, 5th, 7th) and four even positions (2nd, 4th, 6th, 8th).

Step 1: Arrange the odd digits (1, 1, 3, 3) in the four odd positions. The number of ways to arrange these is calculated as 4 factorial divided by (2 factorial times 2 factorial) because the digits 1 and 3 are repeated twice each. Calculation: (4 x 3 x 2 x 1) / (2 x 1 x 2 x 1) = 24 / 4 = 6 ways.

Step 2: Arrange the even digits (2, 2, 4, 4) in the four even positions. The number of ways is calculated the same way because the digits 2 and 4 are also repeated twice each. Calculation: (4 x 3 x 2 x 1) / (2 x 1 x 2 x 1) = 24 / 4 = 6 ways.

Step 3: Total combinations. To get the total number of distinct 8-digit numbers, multiply the arrangements of odd positions by the arrangements of even positions. Total = 6 x 6 = 36.

Therefore, the correct answer is 36.