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Nov 2023 p53 q6
2767
Jai and his wife Kaz are having a party. Jai has invited five friends and each friend will bring his wife. At the beginning of the party, the 12 people will stand in a line for a photograph.
How many different arrangements are there of the 12 people if Jai stands next to Kaz and each friend stands next to his own wife?
How many different arrangements are there of the 12 people if Jai and Kaz occupy the two middle positions in the line, with Jai’s five friends on one side and the five wives of the friends on the other side?
Solution
(i) Consider Jai and Kaz as a single unit or block. Similarly, each friend and his wife can be considered as a block. This results in 6 blocks to arrange. The number of ways to arrange these 6 blocks is given by:
\(6!\)
Within each block, the two people can be arranged in \(2!\) ways. Therefore, the total number of arrangements is:
\(6! \times 2^6 = 720 \times 64 = 46080\)
(ii) Jai and Kaz occupy the two middle positions, so they are fixed. The five friends can be arranged on one side in \(5!\) ways, and the five wives can be arranged on the other side in \(5!\) ways. Therefore, the total number of arrangements is: