What is the probability that celia is first in line and felicity is last in line?

If there is only Celia and Felicity in the line:

There are 2 solutions that Celia and Felicity can be organised in:

Celia 1st in the line, Felicity last in the lane.
Felicity 1st in the line, Celia last in the line.

So, there is a 50% chance that Celia would be 1st in the line and Felicity would be last in the line.


If there is 1 other person in the line:
(The other person will be referred to as Bob)

There are 6 solutions that Bob, Celia and Felicity can be organised in:
(Correct way is in bold)

Bob 1st in line, Celia 2nd in line, Felicity last in line.
Bob 1st in line, Felicity 2nd in line, Celia last in line.
Felicity 1st in line, Bob 2nd in line, Celia last in line.
Felicity 1st in line, Celia 2nd in line, Bob last in line.
Celia 1st in line, Bob 2nd in line, Felicity last in line.
Celia 1st in line, Felicity 2nd in line, Bob last in line.

There is only 1 correct solution, so there is a 16.7% chance that Celia is first in line and Felicity is last in line.
(rounded to 1 decimal place) 

If there are more people, you do what is underneath.

From the 1st section of answers (There were only Celia and Felicity in the line), there were 2 people in the line. The 2nd section of answers showed that 3 people were in the line. So, you times 2 and 3 to make 6. There were 6 solutions in the 2nd section. 

From there, you get the next consecutive number of the people in the line, which is 4 (NOT 7) and times 6 by 4, which is 24. There are 24 solutions to having 4 people in the line. 

Do this again and you get 24 times 5, which is 120. So there are 120 solutions to having 5 people in the line. 

Keep on doing this until you get the number of people in the line you need.

Hopefully this helps!

By the way, this took around 12 minutes to type out.

Edit: Convert the number to a percentage by doing 1/120 times 100.
This gets you 0.83 (rounded to 2 decimal places)
(using the answer of 5 people in the line)