There is a cubical framework (三層田字形, 地下9條柱連接上面天花, 一樓又有三條柱連接上面天花).
1 To go from one end (A) of the frame to the opposite end (B). how many ways (shortest) are there? (ANS=90)
2 There is a trap at the centre of frame (C), find the number of shortest path from A to C ? (ANS=6)
請問可否解釋一下條式？ 很深！不太明，謝！
(a) To go from one end (A) of the frame to the opposite end (B). how many shortest ways are there? ∵ go upward U, rightward R, forward F, at least 2 times. ∴ Number of shortest ways = Number of ways of arranging U, U, R, R, F, F = 6P6 / [(2P2)(2P2)(2P2)] = (2 + 2 + 2)! / (2! 2! 2!) = 90
(bi) There is a trap at the centre of frame (C), find the number of shortest path from (A) to (C) ? because go upward U, rightward R, forward F, at least 1 time. Number of shortest ways = Number of ways of arranging U, R, F = 3P3 = (1 + 1 + 1)! = 6
(bii) Find the probability of reaching (C) on the way from (A) to (B). P (reaching (C) on the way from (A) to (B)) = P (getting one U, one R, one F at front, in arranging U, U, R, R, F, F) = [(6)(3P3)] / 90 = 2/5
