Problem

If radio station call letters must begin with either $\mathrm{K}$ or $\mathrm{W}$ and must include either two or three additional letters, how many different possibilities are there?
There are $\square$ different possibilities.
(Simplify your answer.)

Answer

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Answer

Final Answer: There are \(\boxed{36504}\) different possibilities.

Steps

Step 1 :Break down the problem into two cases: the case where the call letters include two additional letters and the case where the call letters include three additional letters.

Step 2 :For the first case, there are two choices for the first letter (K or W), and for each of the two additional letters, there are 26 choices (the 26 letters of the alphabet). So the total number of possibilities for this case is \(2 \times 26 \times 26 = 1352\).

Step 3 :For the second case, there are again two choices for the first letter, and for each of the three additional letters, there are 26 choices. So the total number of possibilities for this case is \(2 \times 26 \times 26 \times 26 = 35152\).

Step 4 :The total number of possibilities is the sum of the possibilities for these two cases, which is \(1352 + 35152 = 36504\).

Step 5 :Final Answer: There are \(\boxed{36504}\) different possibilities.

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