How many different 4-letter radio station call codes are possible if each code must begin with K or W and no letter can be repeated?

Question

How many different 4-letter radio station call codes are possible if each code must begin with K or W and no letter can be repeated? Example: WABC, KABC, or WABK

Answer 1

Jacqueline:

 

If the code begins with letter K, we have a choice of picking three letters from the remaining 25 without repetition.. This can be done in (25)(24)(23) ways.

 

Similarly, if the code begins with letter W, we have a choice of picking three letters from the remaining 25 without repetition.. This can be done in (25)(24)(23) ways.

Hence the total number of ways of constructing call codes consisting of 4 letters, starting with K or W without repetition of other letters is 2(25)(24)(23).

 

Dr. G.

Answer 2

Many of these types of counting questions can be resolved by thinking about the number of possible symbols in each slot. In this case, since our radio station call code must have 4 letters, we have 4 slots:

 

_ _ _ _ 

 

But instead of filling each slot with a possible letter from A to Z, we're going to fill it in with the number of possible letters that can go in that slot.

 

Since the first letter must be a K or W, the first slot has two options:

 

2 _ _ _ 

 

Now that we've "placed" the first letter, the second slot only has 25 options left, since the second letter must be different from the first, and there are only 26 letters in the alphabet (assuming this radio station is in an English-speaking country). So we have:

 

2 25 _ _ 

 

Now that we've placed the second letter, the third slot only has 24 options, since we've already placed two letters, and the third one has to be different from the first and the second:

 

2 25 24 _

 

Finally, our last slot only has 23 options, since we've already used three different letters for the first three slots:

 

2 25 24 23

 

The final flourish is to simply multiply all of these options together. This gives us:

 

2 x 25 x 24 x 23 = 27,600

 

So the number of different 4-letter radio station call codes satisfying the conditions above is 27,600.

Answer 3

There are 2 choices for the 1st letter (K or W).

Assuming no repetition of letters, once that letter is chosen, there are

 - 25 possible choices for the second letter

- 24 choices for the third letter, and

- 23 choices for the fourth letter

 

So the total number of 4 letter radio stations that must begin with K or W and in which no letter can be repeated is:

2 * 25 * 24 * 23 = 27,600