Question

Six stand up comics, A, B, C, D, E, and F, are to perform on a single evening at a comedy club. The order of performance is determined by random selection. Find the probability that:
1. Comic D will perform first.
2. Comic B will perform fifth and comic A will perform last.
3. The comedians will perform in the following order: D, B, A, C, F, E
4. Combic F or combic E will perform third.

Answer 1

Find the probability that 
1.Comic D will perform first

D _ _ _ _ _
 We can fill the 5 blanks with the 5 letters A, B,C,E,F in 5! = 120 ways
The probability of that is 120 ways out of 720 or 120/720 or 1/6. 

2. Comic B will perform fifth and Comic A will perform last
_ _ _ _ B A
We can fill the 4 blanks with the 4 letters C,D,E, F in 4! = 24 ways.
The probability of that is 24 ways out of 720 or 24/720 or 1/30.

3. The comedians will perform in the following order D, B, A, C, F, E
There is just 1 way that can happen. The probability of that is 1 way out of 720 or 1/720.

4.Comic F or E will perform third.
_ _ F _ _ _
There are 5! = 120 ways to fill those 5 blanks with the letters A,B,C,D, E.
In addition to those, there are another 5! = 120 ways to fill these 5 blanks _ _ E _ _ _ with the letters A,B,C,D,F. That's 2(5!) = 2(120) = 240 ways.
The probability of one of those is 240 ways out of 720 or 240/720 or 1/3.