In Exercises 5–36, express all probabilities as fractions.

Radio Station Call Letters If radio station call letters must begin with either K or W and must include either two or three additional letters, how many different possibilities are there?

Radio station call letters must begin from either K or W and have two or three additional letters.

Thecounting rule involving the multiplication of cases at each step is useful in computingthe total number of possibilities/outcomes of an event.

For the first letter, only K or W are the possibilities.

Thus, the total number of possibilities is two.

Case 1: There are two additional letters.

The number of possibilities for each of the two additional letters is 26.

The total number of possibilities for all three letters is

$2\times 26\times 26=1352$

Case 2: There are three additional letters.

The number of possibilities for each of the three additional letters is 26.

The total number of possibilities for all four letters is

$2\times 26\times 26\times 26=35152$

The total number of possibilities for the letters is the sum of the possibilities of cases 1 and 2.

$$\begin{gathered} \text{Total}\text{possibilities}=\text{1352}+\text{35152} \\ =\text{36504} \end{gathered}$$

Therefore, the total number of different possibilities when the call letter begins with either K or W and contains two or three more additional letters is equal to 36504.