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June 2012 p61 q7
2792
Seven friends together with their respective partners all meet up for a meal. To commemorate the occasion they arrange for a photograph to be taken of all 14 of them standing in a line.
How many different arrangements are there if each friend is standing next to his or her partner? [3]
How many different arrangements are there if the 7 friends all stand together and the 7 partners all stand together? [2]
Solution
(i) Consider each couple as a single unit or 'block'. There are 7 such blocks. These blocks can be arranged in a line in \(7!\) ways. Within each block, the two people can be arranged in \(2!\) ways. Therefore, the total number of arrangements is \(7! \times 2^7 = 645,120\).
(ii) Treat the 7 friends as one group and the 7 partners as another group. These two groups can be arranged in \(2!\) ways. Within each group, the individuals can be arranged in \(7!\) ways. Therefore, the total number of arrangements is \(7! \times 7! \times 2 = 50,803,200\).