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Nov 2015 p62 q4
2781
A group of 8 friends travels to the airport in two taxis, P and Q. Each taxi can take 4 passengers.
Each taxi can take 1 passenger in the front and 3 passengers in the back (see diagram). Mark sits in the front of taxi P and Jon and Sarah sit in the back of taxi P next to each other.
Find the number of different seating arrangements that are now possible for the 8 friends.
Solution
Since Mark is already seated in the front of taxi P, we have 7 friends left to arrange.
Jon and Sarah are sitting in the back of taxi P, so we treat them as a single unit initially. This leaves us with 5 units to arrange: Jon-Sarah, 1 more person in the back of P, and 4 people in taxi Q.
First, choose 1 more person to sit in the back of taxi P from the remaining 5 friends: \\(\binom{5}{1} = 5 \\\).
Now, arrange Jon and Sarah in 2 ways (Jon-Sarah or Sarah-Jon): \\(2! = 2 \\\).
Arrange the 4 people in taxi Q: \\(4! = 24 \\\).
The total number of arrangements is: \\(5 \times 2 \times 24 = 240 \\\).
Since there are 2 possible ways to arrange Jon and Sarah, multiply by 2: \\(240 \times 2 = 480 \\\).