Counting the 10-digit numbers with odd digits
To solve this problem, we need to count the number of 10-digit numbers that can be formed with odd digits, such that no two consecutive digits are the same.

Defining the constraints
In order to create a valid 10-digit number, we need to consider the following constraints:
1. The number should have 10 digits.
2. Each digit should be an odd number (1, 3, 5, 7, or 9).
3. No two consecutive digits should be the same.

Counting the possibilities
To count the number of valid 10-digit numbers, we can use a recursive approach, where we build the number digit by digit while checking the validity at each step.

Initial digit
The first digit of the number can be any of the odd digits (1, 3, 5, 7, or 9). So, there are 5 possibilities for the first digit.

Subsequent digits
For the second digit, we have 4 possibilities since it cannot be the same as the first digit.

For the third digit, we have 4 possibilities as well, since it cannot be the same as the second digit.

Following the same pattern, for the fourth, fifth, sixth, seventh, eighth, ninth, and tenth digits, we also have 4 possibilities each.

Calculating the total number of possibilities
To calculate the total number of possibilities, we need to multiply the number of possibilities for each digit together.

5 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 = 5 * 4^9

Simplifying the expression
To simplify the expression, we can use the fact that any number raised to the power of zero is equal to 1.

5 * 4^9 = 5 * 2^18

Calculating the final answer
To calculate the final answer, we can approximate the value of 2^18, which is approximately 262,144.

5 * 262,144 = 1,310,720

So, the correct answer is 1,310,720, which can be approximated as 5.49.