Let's Play Basketball!

Probability Level 2

There are four basketball players, A, B, C and D practicing. Initially, the ball is with A , the ball is passed from one person to a different person. In how many ways can the ball come back to A , after exactly seven passes?

For example: ( A → D → A → D → C → A → B → A ) (A \rightarrow D \rightarrow A \rightarrow D \rightarrow C \rightarrow A \rightarrow B \rightarrow A) is a way in which the ball can come back to A after 7 passes .

Hint: This question appeared in the Indian National Maths Olympiad 2015 (INMO 2015).

Image Credit: Wikimedia Reisio


The answer is 546.

Arpit MIshra by Arpit MIshra , 21, India

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1 solution

Arpit MIshra
Apr 18, 2015

Answer- Since the ball goes back to one of the 3 persons, we have:

x n + 3 y n = 3 n x_n + 3y_n = 3^n

since there are 3 n 3^n ways of passing the ball in n n passes. Using x n = 3 y n − 1 x_n = 3y_n-1 , we obtain:

x n − 1 + x n = 3 n − 1 x_n-1 + x_n = 3^{n-1} .

with x 1 = 0 x_1 =0 . Thus,

x 7 = 3 6 − x 6 = 3 6 − 3 5 + x 5 = 3 6 − 3 5 + 3 4 − x 4 = x_7 = 3^6 - x_6 = 3^6 - 3^5 + x_5 = 3^6 - 3^5 + 3^4 - x_4 =

3 6 − 3 5 + 3 4 − 3 3 + x 3 = 3 6 − 3 5 + 3 4 − 3 3 + 3 2 − x 2 = 3^6 - 3^5 + 3^4 - 3^3 + x_3 = 3^6 - 3^5 +3^4 - 3^3 + 3^2 -x_2 =

3 6 − 3 5 + 3 4 − 3 3 + 3 2 − 3 3^6 - 3^5 + 3^4 - 3^3 + 3^2 - 3

= ( 2 × 3 5 ) + ( 2 × 3 3 ) + ( 2 × 3 ) (2 \times 3^5) + (2 \times 3^3) + (2 \times 3)

= ( 486 + 54 + 6 ) (486+ 54 + 6)

= 546 \boxed{546}

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