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Five people are passing a ball amongst themselves. The ball starts with Alonzo. Each person who has the ball passes it onto someone else. After the eighth pass, the ball returns to Alonzo. Find the number of possible sequences of passes.

 

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 Jan 2, 2020
 #1
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Draw a circle of the people and label around the circle the letters A,B,C,D, and E (to easily keep track).

We also know that eight passes were completed, and that each person has 6 choices. Find the total.

 Jan 2, 2020
 #2
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Also consider that that Alonzo couldn't have had the ball at the seventh pass (based on the second to last sentence).

Guest Jan 2, 2020
 #4
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Be careful because if Alonzo doesn't get the ball on the seventh throw, then he may or may not get it on the sixth throw and so on. 

Guest Jan 2, 2020
 #3
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There are 4^8 ways of passing the ball, without any restrictions.  After 7 passes, there are 3*4^6 ways the ball could go back to Alonzo (which is not allowed for 8 passes, because the people must be different), so for 8 passes, the number of ways is 4^8 - 3*4^6 = 53248.

 Jan 2, 2020

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