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 letters repeats, ending with a C for the highest key.
a) If five notes are played simultaneously, in how many ways could the notes all be
i)A's? ii)G's? iii)the same letter? iv)different letters?
b) If five keys are played in order, how would your answer in part a) change?
How do you do iv from part a and part b for iv?
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 letters repeats, ending with a C for the highest key.
a) If five notes are played simultaneously, in how many ways could the notes all be
i)A's? ii)G's? iii)the same letter? iv)different letters?
b) If five keys are played in order, how would your answer in part a) change?
How do you do iv from part a and part b for iv?
To calculate the number of ways the notes can be played, we need to consider the following:
a)
i) To find the number of ways to play five A's simultaneously, we can choose all five keys to be A. Since there is only one A key on the piano, the answer is 1.
ii) Similarly, to find the number of ways to play five G's simultaneously, we can choose all five keys to be G. Again, since there is only one G key on the piano, the answer is 1.
iii) To find the number of ways to play five of the same letter, we need to consider the number of choices for a single letter. Since there are 7 distinct letters (A, B, C, D, E, F, G), we have 7 choices for the letter. Therefore, the answer is 7.
iv) To find the number of ways to play five different letters, we need to consider the number of choices for each letter. The first letter can be any of the 7 distinct letters, the second letter can be chosen from the remaining 6 letters, the third letter from the remaining 5 letters, and so on. Therefore, the number of ways to play five different letters is calculated using the concept of permutations. It can be expressed as 7P5, which is equal to 7! / (7 - 5)! = 7! / 2! = 7 * 6 * 5 * 4 * 3 = 2,520.
b)
In part b, where the keys are played in order, the number of ways the notes can be played changes.
a) From part a, for the situation where all five notes are the same letter, the number of ways was found to be 7.
b) However, for the situation where all five letters are different, in part a, it was calculated to be 2,520.
When played in order, we need to account for the arrangements of the keys. Since there are five keys and each can be chosen from a total of 7 distinct letters (A, B, C, D, E, F, G), we can use the concept of permutations. The number of ways to arrange the keys can be calculated as 7P5, which is equal to 7! / (7 - 5)! = 7! / 2! = 7 * 6 * 5 * 4 * 3 = 2,520.
Therefore, both in part a (iv) and part b, the total number of ways to play five different letters is 2,520.