Mathematics, 17.01.2022 04:20 j015

Consider a melody to be 7 notes from a single piano octave, where 2 of the notes are white key notes and 5 are black key notes. Notes can be repeated. How many melodies are possible if the 5 black notes cannot all be "adjacent" in the melody? Adjacent in this context meaning not adjacent on the piano keyboard but occurring without white notes between them. For example, while the pattern WW is not allowed, the pattern WBW is.

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Answer from: starreee

35,829,630 melodies

Step-by-step explanation:

There are 12 half-steps in an octave and therefore 12^7 arrangements of 7 notes if there were no stipulations.

Using complimentary counting, subtract the inadmissible arrangements from 12^7 to get the number of admissible arrangements.

$\displaystyle \_\_ \:B_1\_\_ \:B_2\_\_ \:B_3\_\_ \:B_4\_\_ \:B_5\_\_$

B_1 can be any note, giving us 12 options. Whatever note we choose, $B_2, B_{...}$ must match it, yielding $12\cdot 1\cdot 1\cdot 1\cdot 1=12$ . For the remaining two white key notes, W_1 and W_2 , we have 11 options for each (they can be anything but the note we chose for the black keys).

There are three possible arrangements of white key groups and black key groups that are inadmissible:

$WWBBBBB\\WBBBBBW\\BBBBBWW$

White key notes can be different, so a distinct arrangement of them will be considered a distinct melody. With 11 notes to choose from per white key, the number of ways to inadmissibly arrange the white keys is $\displaystyle\frac{11\cdot 11}{2!}$ .

Therefore, the number of admissible arrangements is:

$\displaystyle 12^7-3\left(\frac{12\cdot 11\cdot 11}{2!}\right)=\boxed{35,829,630}$

Answer from: Quest

answear:

step-by-step explanation: a

Answer from: Quest

answer: a video game has three difficulty levels: easy, medium, and expert. the game's scoring works in such a way that a player has to score a fixed number of points to advance to the next level, and then the scoring starts over. the total score needed to clear all three difficulty levels is 7,500. two times the number of points needed to clear the easy level is 500 more than the number of points needed to clear the medium level. twice the number of points needed to clear the medium difficulty level is 1,500 more than the number of points needed to clear the expert level. if x, y, and z represent the easy, medium, and expert levels, respectively, and a is the coefficient matrix of the system of equations modeling this situation, identify the inverse matrix, a-1, and the solution matrix, x, of the system of equations.

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Consider a melody to be 7 notes from a single piano octave, where 2 of the notes are white key notes...