1. ## Combinations Qusetipm

Just wondered if someone could help me with this...

There are 52 white keys on a piano. The lowest key is A. The keys are designated A,B,C,D,E,F, and G in succession, and then the sequence of lettesr repeats, ending with a C for the highest key.

a) If five notes are played simultaneously, in how many ways could all the notes be:
i) As?
ii) Gs?
iii) the same letter
iv) different letters

b) If the five keys are played in order, hwo would your answers in a) change?

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Okay, so, I've figured out i, ii, and iii for part a, I think..

i) ${8 \choose 5}$ (because A is one of the keys that are repeated at the end)
ii) ${7 \choose 5}$
iii) ${3}{8 \choose 5} + {4}{7 \choose 5}$

And for part b, obviously you just do the math as permutations rather than combinations.

But for Part a, iv), I'm confused about how I should answer. If all they care about is 7 different keys, then your answer would obviously be ${7 \choose 5}$. But, this wouldn't be taking into considering that you could be playing an A in one octave, and a B in another octave. A lower B does not sound the same as a higher B. Is this what the question is indeed looking for? and if so, how would I solve that?

Thanks!

2. You are correct for the questions you answered.
For the last question, you do have to take into account keys being played in different octaves. You have to break up the cases this way:

1. All three of A, B, C are played and two others

2. Two of A, B, C, are played and three others

3. One of A, B, C are played and four others

For 1): $8^3 \cdot 7^2 \cdot {3 \choose 3} \cdot {4 \choose 2}$ cases

For 2): $8^2 \cdot 7^3 \cdot {3 \choose 2} \cdot {4 \choose 3}$ cases

For 3): $8 \cdot 7^4 \cdot {3 \choose 1} \cdot {4 \choose 4}$ cases

The answer is the sum of these three.

### there are 52 white keys on a piano

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