How many options are there for license plate numbers if

VJesus12 wrote: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. 37×260^2
B. 36×260^2
C. 36×270^2
D. 10^2×26^3
E. 10^3×26^2

The OA is the option B.

Why is B? How can I arrive to that number? Could someone help me, please? <i class="em em-disappointed"></i>

Hello Vjesus12.

Let's take a look at your question.

There are two types of making a license plate:

1. 2 digits and 3 letters, in this order. That is to say, the license plate will look like DDLLL.

2. 3 digits and 2 letters, in this order. That is to say, the license plate will look like DDDLL.

The digits can be any number from the set {0,1,2,3,4,5,6,7,8,9}, hence each digit has 10 possible options.

The letters can be any letter of the alphabet, and we are told that the alphabet has 26 letters. Hence, each letter has 26 possible options.

Hence:
1. For the first type of license plate, we have the following number of options:
$$10\cdot10\cdot26\cdot26\cdot26=260\cdot260\cdot26=26\cdot\left(260\right)^2$$

2. For the second type of license plate, we have the following number of options: $$10\cdot10\cdot10\cdot26\cdot26=10\cdot260\cdot260=10\cdot\left(260\right)^2$$

Finally, the total number of options for making license plates is equal to the sum of the number of options for each type, that is to say, $$26\cdot\left(260\right)^2+10\cdot\left(260\right)^2=36\cdot\left(260\right)^2.$$

This is why the correct answer is the option B.

I hope this explanation may help you.

VJesus12 wrote: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. 37×260^2
B. 36×260^2
C. 36×270^2
D. 10^2×26^3
E. 10^3×26^2

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

The number of ways to create the licence 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
.