Question:
Six natives and two foreigners are seated in a compartment of a railway carriage with four seats either side. In how many ways can the passengers seat themselves if
a) the foreigners do not sit opposite each other,
b) the foreigners do not sit next to each other?
a) 6 natives 2 foreigners
the space is like
_ _ _ _
_ _ _ _
the foreigners sit opposite each other
f1 _ _ _ f2 _ _ _
f2 _ _ _ or f1 _ _ _
there are 2 ways of arranging the foreigners ; 2!
the foreigners can sit opposite each other in four ways
the six natives can be arrange in 6! ways to fill the six spaces left
thus the answer is 2! x 4 x 6! = 34560
b)
I am having problems with this one
my idea is to find out the number of ways in which the two foreigners are seated together and then subtract this from the total number of ways of them seating with no restriction.
Suppose the two foreigners are considered as a single entity then there are 7 objects. therefore ti can be arranged in 7! ways and the two foreigners can be arranged in 2! ways.
thus; 2! x 7! ways for the two foreigners can sit together
now to find the real answer we do
8! - (2! x 7!) = 30240
but the book gives an answer of 31680
Please help me what did i do wrong for part b