Combinatorics: License Plate

Jan 2014
Digits include 0, 1, 2, ..., 9.

How many license plates can be made out of 3 letters and 3 digits if the first and the last symbol must be a digit?

These are my thoughts:

At the first and last positions we must have a digit, so there are 10 possibilities in each of those slots.
For the remaining four slots we must have 3 letters and 1 digit, but the order does not matter.
So in 3 of these slots we have 26 possibilities, and in one of these slots we have 10 possibilities.
In addition, there are 4 different slots that we could put the digit in.
So, the number of ways to construct such a license plate would be: 10 * 26 * 26 * 26 * 10 * 4 * 10 = 70304000.

Is this true?


MHF Hall of Honor
May 2006
Lexington, MA (USA)
Hello, Convrgx!

I agree with your answer.


MHF Helper
Nov 2013
This is correct. A slightly more concise way of figuring out the middle part is that you have to have 3 letters and 1 digit in 4 slots.

There are $\begin{pmatrix}4 \\ 3\end{pmatrix} = 4$ ways of arranging this as you noted by intuition.
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