1. ## Permutation Question

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 _ _ _

or

f2 _ _ _
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 8!- (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. the two foreigners can be arranged in 2! ways. Six natives can be arranged in 6! ways

thus; 2! x 6! ways for the two foreigners can sit together

now to find the real answer we do

8! - (2! x 6!) = 38880

but the book gives an answer of 31680

So we arrive at $8! - 12\cdot 6! = 6!(56-12) = 31680$ arrangements