Question

How many different 4-letter radio station call letters can be made
a. if the first letter must be C, W, P, or N and no letter may be repeated?
b. if repeats are allowed (but the first letter is C, W, P, or N)?
c. How many of the 4-letter radio station call letters (starting with C, W, P or N) have no repeats and end with the letter R?

a. If the first letter must be C, W, P or N, and all the letters in the radio station call letters must be different, then there are _________ 4-letter radio station call letters.

b. If the first letter must be C, W, P or N, and letters can be repeated, then there are __________ radio station call letters.

c. If the first letter must be C, W P or N, and the last letter must be R and all the letters in the radio station call letters must be different, then there are ________ 4-letter words.

Answer 1

a) Given that the first letter must be C, W, P or N there are 4 ways to fill the first letter. After filling the first letter, the number of ways to fill the remaining 3 letters from 25 remaining letters is computed as the permutation of 25 letters taken 3 at a time. This is computed here as:

= 4*25*24*23

= 55200

therefore there are 55200 ways to form the 4 letter radio station letter here.

b) Given repeatition is allowed here, we have no restriction on the last 3 characters, each one of them could be chosen from the available 26 letters. Therefore total number of ways to form the 4 - letter code is computed here as:

= 4*26*26*26

= 70304

Therefore there are 70304 ways to form the 4 letter radio station letter here.

c) Given that last letter must be R and repetition is not allowed, we first choose the first letter in 4 ways, after which we are left with 26 - 2 = 24 letters. The number of ways to fill the middle 2 letters is computed as the permutation of 24 letters taken 2 at a time. Therefore total number of ways here is computed as:

= 4*24*23

= 2208

Therefore there are 2208 ways to form the 4 letter radio station letter here.