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# Counting

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In a certain region all license plates are composed of three letters followed by three numbers, or three numbers followed by three letters. The only restriction is that zero can never be the first of the three numbers. Find how many license plates are possible.

Dec 30, 2021

#1
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If it is composed of 3 letters, then 3 numbers, there is 26^3 * 10^3 ways to create a license plate.

If the plate starts with 3 numbers however, there are 9 choices for the first number (1 through 9), and 10 choices for the other 2.  There are 26^3 ways to choose 3 letters after that, meaning that there are 9* 10^2 * 26^3 ways to arrange a license plate this way.

In total, there are 26^3 * 10^3 +9 * 10^2 * 26^3, or 33394400

Dec 30, 2021
#2
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Slightly diffeent answer.... the number cannot begin with a zero

so you have    9 * 10 * 10 = 900 choices for the number

you have 26 * 26 * 26 = 17576 choices for the letter portion

and you can have the numbers first or the letters first = 2 choices

900 * 17576 * 2 = 31,636,800 plates

Dec 30, 2021