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Nov 2021 p51 q5
2813
Raman and Sanjay are members of a quiz team which has 9 members in total. Two photographs of the quiz team are to be taken.
For the first photograph, the 9 members will stand in a line.
(a) How many different arrangements of the 9 members are possible in which Raman will be at the centre of the line?
(b) How many different arrangements of the 9 members are possible in which Raman and Sanjay are not next to each other?
Solution
(a) To find the number of arrangements with Raman at the center, fix Raman in the center position. There are 8 remaining members to arrange around him. The number of ways to arrange these 8 members is given by the factorial of 8, which is:
\(8! = 40,320\)
(b) To find the number of arrangements where Raman and Sanjay are not next to each other, first calculate the total number of arrangements of the 9 members, which is \(9!\). Then, calculate the number of arrangements where Raman and Sanjay are together. Treat Raman and Sanjay as a single unit, so there are 8 units to arrange. The number of ways to arrange these 8 units is \(8!\), and within the unit, Raman and Sanjay can be arranged in \(2!\) ways. Thus, the number of arrangements where they are together is \(8! \times 2!\).
The number of arrangements where they are not together is:
