An octave contains twelve distinct notes (on a piano, five black keys and seven white keys). how many different eight-note melodies within a single octave can be written if the black keys and white keys need to alternate?

Ok, so you have 8 white notes and 5 black.

You want to write an 8 note melody, with te only restriction that you need to alternate the colors.

so if we start with the white keys you have

8 possibilities, then 5, then 8, then five, and so on.

so you have

and also you can start with a black key, that will give you the same amount of combinations.

so total number is 2* = 512000

the answer is 5636.7