For many years, a state used 5 letters followed by 1 digit on its automobile license plates. Complete parts (a) through (c) below. (a) How many different license plates are possible with this arrangement? (Use scientific notation. Use the multiplication symbol in the math palette as needed. Round to three decimal places as needed.) (b) When the state ran out of new numbers, the order was reversed to 1 digit followed by 5 letters. How many new license plate numbers were then possible? (a)

We start by noting that each license plate consists of 5 letters followed by 1 digit. 1. For each letter position there are \(26\) possible letters, so for 5 letters there are \[ 26^5 \] possibilities. 2. For the digit position there are \(10\) possible digits, so the digit contributes a factor of \[ 10. \] 3. Multiplying the number of choices for the letters and the digit, the total number of different license plates is \[ 26^5 \times 10. \] Now we compute \(26^5\): \[ 26^2 = 676, \quad 26^3 = 26 \times 676 = 17576, \] \[ 26^4 = 26 \times 17576 = 456976, \quad 26^5 = 26 \times 456976 = 11881376. \] Multiplying by \(10\): \[ 11881376 \times 10 = 118813760. \] In scientific notation this is approximately \[ 1.188 \times 10^8. \] Thus, the total number of license plates possible is \[ \boxed{1.188 \times 10^8}. \]