Question

Suppose that in a certain state, all automobile license plates have four letters of the 26 English alphabet letters followed by three digits from 0,1,2,3,4,5,6,7,8,9. For example, WCRY-122. How many license plates are possible that begins with A and ends with 1 with NO repetition

Answer 1

Answer:

$993,\!600$.

Step-by-step explanation:

There is one way to choose the first character (has to be an $\verb!A!$.)

There is one way to choose the last character (has to be a $\verb!1!$.)

Since repetition among the letters is not allowed and the letter $\verb!A!$ was already reserved for the first character, there would be $(26 - 1) = 25$ English alphabet letters available for the second character.

Likewise, with the first and second characters chosen, the third character could be chosen from $(26 - 2) = 24$ letters. There would be $(26 - 3) = 23$ choices for the fourth letter.

Repetition of the digits is not allowed, either. With the digit $\verb!1!$ already reserved for the last character, there would be $(10 - 1) = 9$ ways to choose the fifth character (a digit.) There would then be $(10 - 2) = 8$ ways to choose the sixth character (also a digit.)

Overall, the number of unique combinations would be:

$\begin{aligned}& 1 \times 1 \times (25 \times 24 \times 23) \times (9 \times 8) \\ =& \; 993,\!600 \end{aligned}$.