This problem involves analyzing how the positions of digits in a number change when specific parts of the number are rearranged.

Understanding the Number and Halves

The given number is $2451379638$. It has 10 digits.

• The number can be split into two halves:
• First half: $24513$ (the first 5 digits)
• Second half: $79638$ (the last 5 digits)

Rearranging the Halves

The digits within each half need to be arranged in ascending order separately.

• First half ($24513$): The digits are $2, 4, 5, 1, 3$. Arranging these in ascending order gives $1, 2, 3, 4, 5$.
• Second half ($79638$): The digits are $7, 9, 6, 3, 8$. Arranging these in ascending order gives $3, 6, 7, 8, 9$.

Forming the New Number

Now, combine the sorted digits from both halves to form the new number:

• Sorted first half: $12345$
• Sorted second half: $36789$
• The new number is $1234536789$.

Comparing Digit Positions

Let's compare the original number with the new number to see how many digits remain in their original positions.

Counting Unchanged Digits

By comparing the digits at each position, we find that none of the digits in the original number $2451379638$ remain in the same position in the new number $1234536789$.

• Therefore, the number of digits whose positions remain the same is 0.

Final Answer Calculation

The question asks for the count of digits that remain in the same position after sorting the halves. Based on the comparison:

• Number of digits unchanged = 0

This corresponds to the option 'Zero'.