Question

How many different 6-letter radio station call letters can be made a. if the first letter must be G or X and no letter may be repeated? b. if repeats are allowed (but the first letter is G or X)? c. How many of the 6-letter radio station call letters (starting with G or X) have no repeats and end with the letter R?

Answer 1

We need 6 letter radio station call letters

a) 1st letter is choosed from letters G or X as 2c1 =2

No letters are repeated. So the next 5 letters chooses from the remaining (26-1=25) 25 letters as

= 25c1 × 24c1× 23c1 × 22c1×21c1

= 25×24×23×22×21

So 6 letters formed as = 2 × 25×24×23×22×21

= 12,751,200 ways

b) 1st letter is choosen from G or X as 2c1 ways

And repetition is allowed then the next 5 letters chooses from 26 letters as 26c1×26c1×26c1×26c1×26c1

So 6 letters formed as

= 2× 26×26×26×26×26

=23,762,752 ways

C) 1st letter is choose from G or X as 2c1 ways and no repeat and the last letter is must R it chooses as 1 way.  then the remaining  4 letters chooses from 24 letters ( not G or X and R)

as 24c1×23c1×22c1×21c1

= 24×23×22×21ways

Then the 6 letters are

= 2× 24×23×22×21×1

= 510,048 ways