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June 2015 p63 q7
2782
Rachel has 3 types of ornament. She has 6 different wooden animals, 4 different sea-shells and 3 different pottery ducks.
Rachel displays 10 of the 13 ornaments in a row on her window-sill. Find the number of different arrangements that are possible if
(ii) she has a duck at each end of the row and no ducks anywhere else,
(iii) she has a duck at each end of the row and wooden animals and sea-shells are placed alternately in the positions in between.
Solution
(ii) To have a duck at each end, choose 2 ducks from the 3 available. This can be done in \(^3P_2\) ways. The remaining 8 positions can be filled with the 10 remaining ornaments (6 wooden animals and 4 sea-shells) in \(^10P_8\) ways. Therefore, the total number of arrangements is \(^3P_2 \times ^{10}P_8 = 10886400\).
(iii) Place a duck at each end, which can be done in \(^3P_2 = 6\) ways. The remaining 8 positions must alternate between wooden animals and sea-shells. Arrange the 4 sea-shells in \(^4P_4 = 24\) ways and the 4 wooden animals in \(^6P_4 = 360\) ways. Since the arrangement must alternate, swap the positions of sea-shells and wooden animals in 2 ways. Therefore, the total number of arrangements is \(6 \times 24 \times 360 \times 2 = 103680\).