Question

If 2 people out of 7 refuse to sit next to each other in a row of 7 seats, how many ways are possible?

Answer 1

Hint: Find total possibility and subtract it from the ways the arrangement is not acceptable. First, we find out the total possible ways of seating all 7 people in the row and then we find to consider the undesirable pair by seating one of them with the other 6 people and then finding the possibility of not pairing them together and then subtract them from the total possible arrangements of the 7 people.

Complete step by step solution:
First, we are going to find the total possible way of arrangement of seating that can be done to a row of 7 seats.
Since, they are no restriction, a single person has a possibility of sitting in all the non-occupied seats with no restriction. So, the total ways of arrangements are $7!$ ways.
So,
$
\Rightarrow 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \\
\Rightarrow 5040 \\
$
The total ways of arranging are $5040$ ways of seating 7 people in the row with no restriction.
Next, we have to find the total ways in which there are restrictions and the ways of arrangement are not at all accepted.
To find the number of these ways that are unacceptable because they result in the particular pair being sat next to one another, we can think of seating 6 people in a row of 6 seats, excluding one person from the particular pair, then suppose they arrive and squeeze an extra seat into the row one side or the other of the other member of the pair.
Then the ways are
$
\Rightarrow 2\times \,\,6! = 2(6 \times 5 \times 4 \times 3 \times 2 \times 1) \\
  \Rightarrow 1440 \\
$
There are a total $1440$ ways of arrangements which are not acceptable.
Now, if we subtract them from the total ways of arrangement, we get the required arrangement.
So, the total ways of acceptable arrangement with the restriction are
$
  \Rightarrow 5040 - 1440 \\
  \Rightarrow 3600 \\
$
Which is a total of $3600$ ways.
Hence the correct answer is 3600 ways.

Note: While we make the arrangement with seating, we have to make sure the pair is not formed, by eliminating one of them from the initial arrangement and then later, consider them in the arrangement, to ensure that the restriction is followed.