An octave contains twelve different musical notes (in Western music). How many different eight note melodies can be constructed from these twelve notes if:
(a) no note can be used more than once?
(b) any note can be used as often as you please?
================================
Ans for a) a permutation
12P8 =19,958,400
b) a combination
12C8= 495
Is this correct?
Yes, your answers are correct!
To calculate the number of different eight-note melodies, we can use permutations and combinations.
For part (a), where no note can be used more than once, we need to use permutations. Since there are 12 different notes to choose from for the first position, then 11 for the second position, and so on until 5 for the last position (since we need 8 notes in total), we can calculate it as:
12P8 = 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 = 19,958,400 different melodies.
For part (b), where any note can be used as often as you please, we can use combinations. We need to choose 8 notes out of the 12 available notes, and order does not matter. Therefore, we can calculate it as:
12C8 = 12! / (8! * (12-8)!) = 12! / (8! * 4!) = 495 different melodies.
So, your answers for part (a) and part (b) are indeed 19,958,400 and 495 respectively. Well done!