Can you solve it?#4
There are four basket-ball players A,B,C,D. Initially, the ball is with A. The ball is always passed from one person to a different person. In how many ways can the ball come back to A after seven passes? (For example A → C → B → D → A → B → C → A and A →D→A→D→C→A→B→Aaretwowaysinwhichtheball can come back to A after seven passes.)
The answer is 546.
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1 solution
Let xn be the number of ways in which A can get back the ball after n passes. Let yn be the number of ways in which the ball goes back to a fixed person other than A after n passes. Then xn = 3yn−1, and yn = xn−1 +2yn−1. We also have x1 = 0, x2 = 3, y1 = 1 and y2 = 2. Eliminating yn and yn−1, we get xn+1 = 3xn−1 +2xn. Thus x3 = 3x1+2x2 =2×3=6; x4 = 3x2+2x3 =(3×3)+(2×6)=9+12=21; x5 = 3x3+2x4 =(3×6)+(2×21)=18+42=60; x6 = 3x4+2x5 =(3×21)+(2×60)=63+120=183; x7 = 3x5+2x6 =(3×60)+(2×183)=180+366 =546.