Question: How many different license plates are possible if each contains 2 letters (out of the alphabet’s 26…

How many different license plates are possible if each contains 2 letters (out of the alphabet’s 26 letters) followed by 5 digits (from 0 to 9)? How many of these license plates contain no repeated letters and no repeated digits?

There are [ ] different possible license plates. (Simplify your answer.)

Solution

To solve the problem of how many different license plates are possible with 2 letters followed by 5 digits, where no repeated letters or digits are allowed, follow these steps: 1. Calculate the number of ways to choose the letters: There are 26 letters in the alphabet. For the first letter, you have 26 choices. For the second letter, you have 25 choices (since it cannot be the same as the first letter). \[ 26 \times 25 \] 2. Calculate the number of ways to choose the digits: Digits range from 0 to 9, totaling 10 digits. For the first digit, you have 10 choices. For each subsequent digit, the choices decrease by one as no repeats are allowed. \[ 10 \times 9 \times 8 \times 7 \times 6 \] 3. Calculate the total number of possible license plates: Multiply the number of ways to choose the letters by the number of ways to choose the digits. \[ (26 \times 25) \times (10 \times 9 \times 8 \times 7 \times 6) \] 4. Simplify the result: \[ 650 \times 30240 = 19,656,000 \] There are 19,656,000 different possible license plates.