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Counting - Addition and Multiplication

For COMPETITION
Number of Total Problems: 1.
FOR PRINT ::: (Book)

Problem Num : 1

From : AMC10

Type:

Section:Counting

Theme:
Adjustment# : 0

Difficulty: 1

'
A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible?
$mathrm{(A) } 10^4 imes 26^2qquadmathrm{(B) } 10^3 imes 26^3qquadmathrm{(C) } 5 imes 10^4 imes 26^2qquadmathr...$
'

Category Addition and Multiplication

Analysis

Solution/Answer
There are $10cdot10cdot10cdot10 = 10^4$ ways to choose 4 digits.
There are $26 cdot 26 = 26^2$ ways to choose the 2 letters.
For the letters to be next to each other, they can be the 1st and 2nd, 2nd and 3rd, 3rd and 4th, 4th and 5th, or the 5th and 6th characters. So, there are $6 - 1 = 5$ choices for the position of the letters.
Therefore, the number of distinct license plates is $5 imes 10^4 imes 26^2 Longrightarrow mathrm{C}$ .

Answer:

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