# Find the number of ways of arranging n people in a straight line, if two particular people must always be separated..?

$\text{Ways" = 2x((n-2)!), "where } x = {\sum}_{y = 1}^{n - 2} y$

#### Explanation:

Let's work a few examples with a growing value of $n$ and see if we can find a pattern. Let's assign A and Z to the two people who must be separated.

When $n = 3$, there are 2 ways we can sit the people - AXZ and ZXA. Ordinarily, we'd be able to sit 3 people in 3! = 6 ways.

When $n = 4$, we can sit A and Z in seats 1, 3; 1, 4; and 2, 4 - which is 6 different ways. The other two people can sit in 2! = 2 ways for each of the 6 ways we can seat A and Z, which is $6 \times 2 = 12$ ways. Ordinarily, we could seat four people 4! = 24 ways.

When $n = 5$, we can seat A and Z in seats 1, 3; 1, 4; 1, 5; 2, 4; 2, 5; 3, 5 - which is 12 different ways. The remaining three people can sit in 3! = 6 different ways, giving $12 \times 6 = 72$ ways. Ordinarily, we'd seat 5 people in 5! = 120 ways.

When $n = 6$, we can seat A and Z in seats 1, 3; 1, 4; 1, 5; 1, 6; 2, 4; 2, 5; 2, 6; 3, 5; 3, 6; 4, 6 - which is 20 ways. The remaining four people can sit in 4! = 24 different ways, giving $20 \times 24 = 480$ ways. Ordinarily, we'd seat 6 people in 6! = 720 ways.

What can we find from the above?

• We can express the ways to seat the people other than A and Z as (n-2)!

• We can express the ways to seat A and Z as $2 x$. It's the $x$ that is the thing we need to express that is the problem.

When:
- $n = 3 , x = 1$
- $n = 4 , x = 3$
- $n = 5 , x = 6$
- $n = 6 , x = 10$.

And so we can express $x$ as the sum of the natural numbers from 1 to $\left(n - 2\right)$:

$x = {\sum}_{y = 1}^{n - 2} y$

And that is the expression we can use to answer our question:

$\text{Ways" = 2x((n-2)!), "where } x = {\sum}_{y = 1}^{n - 2} y$

Let's give this a quick test. For $n = 6$, we have:

2x((6-2)!), x=sum_(y=1)^(n-2)y=1+2+3+4=10

2(10)(4!)=20xx24=480

For fun, let's try $n = 10$:

2x((10-2)!), x=sum_(y=1)^(n-2)y=1+2+3+4+5+6+7+8=36

2(36)(8!)=72xx40320=2,903,040