Question

license plates in a particular state display 2 letters followed by 4 numbers. how many different license plates can be manufactured for this state?
there are \\(\square\\) different license plates that can be manufactured for this state.
(simplify your answer. type an integer or a fraction.)

Explanation:

Step1: Calculate the number of possibilities for letters

There are 26 letters in the alphabet. For the two - letter part of the license plate, by the multiplication principle (if one event has \(m\) outcomes and another independent event has \(n\) outcomes, the total number of outcomes for the two events together is \(m\times n\)), the number of ways to choose 2 letters is \(26\times26\) since for the first letter we have 26 choices and for the second letter we also have 26 choices.

$$26\times26=26^{2}=676$$

Step2: Calculate the number of possibilities for numbers

There are 10 digits (\(0 - 9\)). For the four - number part of the license plate, using the multiplication principle, the number of ways to choose 4 numbers is \(10\times10\times10\times10\) (since for each of the four number positions, we have 10 choices).

$$10\times10\times10\times10 = 10^{4}=10000$$

Step3: Calculate the total number of license plates

By the multiplication principle (combining the letter and number parts), the total number of license plates is the product of the number of letter combinations and the number of number combinations.

$$N=26^{2}\times10^{4}$$

$$N = 676\times10000=6760000$$

Answer:

\(6760000\)