An octave contains 12 distinct notes(on a piano, five black keys and seven white keys). How many different eight notes melodies within a single octave can be written if the black keys and white keys need to alternate.
Attempt: Let A1*A2*A3*A4*A5*A6*A7*A8 be a different eight note melody. Then A1 can start with either black key or white key. Then an octave has 8 keys to play.I know the total number of all possible eight note melodies have to be the sum of the withe and black different eight note melodies. However, can someone please help me at least put the alternating octave when it starts with black keys? Thank you very much.
If I understood you correctly:
1. Notes can repeat:
If you start with "white", then it's seven options times five options, etc: $$7 \times 5 \times 7 \times 5 \times 7 \times 5 \times 7 \times 5 = 7^4 5^4$$
If you start with "black", then it's five options times seven options, etc:
$$ 5 \times 7 \times 5 \times 7 \times 5 \times 7 \times 5 \times 7 = 7^4 5^4$$
Total is: $$2 \times7^4 5^4$$
2. Notes cannot repeat:
If you start with "white", then it's seven options times five options, etc: $$7 \times 5 \times 6 \times 4 \times 5 \times 3 \times 4 \times 2 = 20 \times 7!$$
If you start with "black", then it's five options times seven options, etc:
$$ 5 \times 7 \times 4 \times 6 \times 3 \times 5 \times 2 \times 4 = 20 \times 7!$$
Total is: $$5 \times 8!$$