SHOW WORK! SHOW STEPS! A piano has 88 keys. How many ways can three keys be play

Question

SHOW WORK! SHOW STEPS!

A piano has 88 keys. How many ways can three keys be played in succession? How many ways can three different keys be played in succession? How many ways can three different keys be played simultaneously? How many ways can seven keys be played in succession? How many ways can seven different keys be played in succession? How many ways can seven different keys be played simultaneously? Assume at least one key is played. How many ways can up to seven keys be played in succession? How many ways can up to seven different keys be played in succession? How many ways can up to seven different keys be played simultaneously?

Explanation / Answer

(a)(i) 88*88*88

(ii)88P3 (number of permuatations of 88 objects taken 3 at a time)

(iii) 88C3 (number of ways 3 keys can be selected from 88 keys,arrangement doesnot matter as keys are played simultaneously)

(b)(i) 88^7

(ii) 88P7

(iii)88C7

(c)(i) 88 + 88^2 + ... + 88^7 (number of ways i keys can be placed in succession = 88^i,here 1<=i<=7)

(ii) 88P1 + 88P2 + ... + 88P7

(88Pi is the number of ways i different keys can be played in succession,we are taking permutations since order is important)

(iii) 88C1 + 88C2 + ... +88C7

(here order is not important, all keys are played simultaneously)

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