How many options are there for license plate number if each

BTGmoderatorLU wrote:Source: Economist GMAT

How many options are there for license plate numbers if each license plate can include 2 digits and 3 letters (in that order), or 3 digits and 2 letters (in that order)? (Note: there are 26 letters in the alphabet)

A. \(360\cdot 260^2\)
B. \(36\cdot 260^2\)
C. \(36\cdot 270^2\)
D. \(10^2 \cdot 26^3\)
E. \(10^3 \cdot 26^2\)

The OA is B

Number of ways each license plate can include 2 digits and 3 letters (in that order) = 10^2 * 26^3; (there are 10 digits, incl 0; note that digits and alphabets can repeat)

Number of ways each license plate can include 3 digits and 2 letters (in that order) = 10^3 * 26^2

Total number of ways = 10^2 * 26^3 + 10^3 * 26^2 = 10^2 * 26^2(26 + 10) = 36 * 10^2 * 26^2 = 36 * 260^2

The correct answer: B

Hope this helps!

-Jay
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BTGmoderatorLU wrote:Source: Economist GMAT

How many options are there for license plate numbers if each license plate can include 2 digits and 3 letters (in that order), or 3 digits and 2 letters (in that order)? (Note: there are 26 letters in the alphabet)

A. \(360\cdot 260^2\)
B. \(36\cdot 260^2\)
C. \(36\cdot 270^2\)
D. \(10^2 \cdot 26^3\)
E. \(10^3 \cdot 26^2\)

The OA is B

The number of ways to create the license plate with 2 digits followed by 3 letters is 10^2 x 26^3.

The number of ways to create the license plate with 3 digits followed by 2 letters is 10^3 x 26^2.

So we have:

10^2 x 26^3 + 10^3 x 26^2

This expression is not one of the answer choices, so we simplify it by factoring the common (10^2) and (26^2) from each term. Thus, we have:

10^2 x 26^2 x (26 + 10) = (26^2)(10^2)(36) = (260^2)(36)

Answer: B