1. **Problem Statement:** We need to find the number of possible special plate numbers consisting of 3 letters followed by 2 digits. 2. **Given:** - Plate format: 3 letters + 2 digits - Letters: English alphabet (26 letters) - Digits: 0 to 9 (10 digits) 3. **Method:** Use the multiplication principle of counting, which states that if there are $n$ ways to do one thing and $m$ ways to do another, then there are $n \times m$ ways to do both. --- ### a. Letters and digits can be repeated - Number of ways to choose each letter = 26 (since repetition allowed) - Number of ways to choose each digit = 10 (since repetition allowed) Number of possible plates: $$ 26 \times 26 \times 26 \times 10 \times 10 = 26^3 \times 10^2 $$ Calculate: $$ 26^3 = 26 \times 26 \times 26 = 17576 $$ $$ 10^2 = 10 \times 10 = 100 $$ So, $$ 17576 \times 100 = 1,757,600 $$ --- ### b. Letters and digits cannot be repeated - Number of ways to choose first letter = 26 - Number of ways to choose second letter = 25 (one letter used) - Number of ways to choose third letter = 24 (two letters used) - Number of ways to choose first digit = 10 - Number of ways to choose second digit = 9 (one digit used) Number of possible plates: $$ 26 \times 25 \times 24 \times 10 \times 9 $$ Calculate stepwise: $$ 26 \times 25 = 650 $$ $$ 650 \times 24 = 15600 $$ $$ 15600 \times 10 = 156000 $$ $$ 156000 \times 9 = 1,404,000 $$ --- **Final answers:** - a. $1,757,600$ possible plates - b. $1,404,000$ possible plates This shows how the multiplication principle helps count the total number of combinations when repetition is allowed or not allowed.