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A standard Missouri state license plate consists of a sequence of two letters, one digit, one letter, and one digit. How many such license plates can be made?

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    $\begingroup$ Start small. How many sequences of $2$ not necessarily distinct letters are there? $\endgroup$ – André Nicolas Mar 14 '13 at 1:12
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L-L-D-L-D:

  • L = letter $\in \{a, b, c, d, ..., x, y, z\}$; $\;26$ letters in the alphabet
  • D = Digit $\in \{0, 1, 2, ..., 8, 9\}$; $\;10$ possible digits to choose from

You have:

$\quad$ ___options for the first letter

$\times $ ___options for the second letter (not necessarily distinct from the first letter)

$\times$ ___options for the first digit

$\times$ ___options for the last letter, (not necessarily distinct from the first or second letter)

$\times$ ___options for the last digit...(not necessarily distinct from the first digit)

= total number of possible license plates that can be produced in Missouri.

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Hint: Total number of arrangements = the product of the number of arrangements at each position.

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$L$ is a letter in $\{a,b,c,d,\dotsc,x,y,z\}$; $26$ letters in the alphabet.

$D$ is a Digit in $\{0,1,2,\dotsc,8,9\}$; $10$ possible digits to choose from

$$26\cdot26\cdot10\cdot26\cdot10 = 1757600$$

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