A four-digit number must be constructed. This number must begin with the digit \( 4 \) and terminate with either \( 0 \) or \( 5 \). The initial and terminal digits are predetermined, whereas the intermediate digits are variable.
The initial digit is fixed at \( 4 \), allowing only one selection.
Each of the second and third digits can be any integer from \( 0 \) to \( 9 \), resulting in ten possibilities for each.
The terminal digit can be either \( 0 \) or \( 5 \), providing two options.
To ascertain the total count of permissible four-digit numbers, the number of options for each digit is multiplied:
\[1 \times 10 \times 10 \times 2 = 200.\] Final Answer:\[\boxed{200}\]