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?

The total different possible combinations of letters for the radio stations with 3 or 4 letters, such that they start with k or w, is 36504.Further explanationTo answer this question we need to find all possible combination of letters with a length of 3 or 4, regarded that those combinations start with k or w. The reason why we need to find all combinations with a length of 3 or 4 is because we must start with k or w, and then add either 2 or 3 letters. To properly understand this question, first we need to know how many letters we can deal with, if we are dealing with the English alphabet, then we have a total of 26 letters we can use (from a to z).Let's start simple with an example to understand the concept. Suppose we wish to compute all possible combination of letters of length 2, such that the first letter is k. The answer to this would be 26 since we are counting words like ka, kb, kc, ..., kz, which sums up to 26 (26 possibilities of the second letter, for just 1 possibility of the first letter).Now let's add a new letter to our previous example, so suppose we wish to compute all possible combination of letters of length 3, such that the first letter is k. The answer to this would be [tex]26^2[/tex], since we are counting words like kaa, kab, kac, ... kaz, kba, kbb, kbc, ..., kbz, ..., kza, kzb, kzc, ... kzz, which sums up to [tex]26^2[/tex] (26 possibilities of the third letter, for every 26 possibilities of the second letter, for just 1 possibility of the first letter).From the previous example, it's straightforward to see that the total number of combination of letters of length 4 such that the first letter is k, is [tex]26^3[/tex]. This gives us all which is necessary to solve our problem.So, if the first letter is k in our combination, the total number of possible combinations of length 3 and 4 will be [tex]26^2 + 26^3[/tex], which is equal to 18252. In case, the first letter of the combination is w, we have the same result, 18252. Therefor the answer to our question will be [tex]18252 + 18252 = 36504[/tex].Learn moreOrdered combinations: combinations: problem on combinations: , possibilities, combinations.