License plates on cars in a certain state consist of 3 letters

License plates on cars in a certain state consist of 3 letters taken from the 26 letters, A through Z, followed by 3 digits taken from the 10 digits, 0 through 9. Which of the following expressions gives the number of distinct license plates that are possible given that repetition of both letters and digits is allowed?

  1. 103 · 263
  2. (10 + 26)3
  3. 2(26!)3(10!)3
  4. (3 + 3)26 + 10
  5. (26! · 10!)3 + (26! · 10!)3

Answer

The correct answer is A.

The Fundamental Counting Principle states that if event A can occur in m ways, and for each of these m ways event B can occur in n ways, there are exactly m × n ways both events can occur together.

In making a license plate for this state, we have to make six decisions. We have to choose a letter, another letter, another letter, a digit, another digit, and one more digit. There are 26 ways to choose a letter. Because duplications are allowed, there are 26 ways to choose the second letter, and 26 ways to choose the third letter. Similarly, there are 10 ways to choose the first digit, 10 ways to choose the second digit, and, finally, 10 ways to choose the third digit.

Using the Fundamental Counting Principle, there will be (26)(26)(26)(10)(10)(10) = 103 · 263 distinct license plates in this state.