Probability question

[299678]

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Q. a theater has 5 doors, in how many ways can an actor enter one door and leave in a different door
-> 20 ways

Q. a piece of paper has 10 distinct dots drawn on it. If a pair of dots determines a line, how many different ways are there to draw a line on this paper
-> 40

the first question is permutation and the second question is combination. Could you explain how to differentiate between permutation and combination on above examples?

 

[herkulease]

With permutation, order matters. With combination order does not matter.

With the 2nd case. I get the answer of 45 lines. Not sure how they got 40.

10!/2!(8!)

 

[SmilezZz]

Q. a theater has 5 doors, in how many ways can an actor enter one door and leave in a different door
-> 20 ways

The actor enters one door and leaves through a different door, so he's only going through a total of 2 doors.
For the 1st door, he can choose from a total of 5 different doors. He has to come out of a different door, so now he can only choose from a total of 4 doors.
5 x 4 = 20

If you like, you can write out all the different permutations.
Label the doors A thru E.
If the actor enters through door A, he can leave through B, C, D, or E.
AB
AC
AD
AE

Repeat the same process, except this time with the actor entering through door B.
BA
BC
BD
BE

Repeat the process so the actor enters through each door, and you will have a total of 20.

 

[UCB05]

In the first case, the wording makes it sound like order matters. He walks into one door and walks out another. This would be different than if he did it backwards, into the second and out the first, because the only stipulation is that he walks in and out of different doors. If the question were "how many pairs of doors can he walk in and out of?" then it would sound more like a combination question, since going out one way or backwards represents the same pair of doors.

In the second question, you're only dealing with the line that is drawn from the two points. Order doesn't matter, as p1>p2 and p2>p1 both make the same line. I'm getting the same answer as Herk, 45.

 

[299678]

Thank you guys all ~

 

[Kittenz]

I've watched chads videos on this, and yet I can't seem to always figure out whether order matters or not. The wording specifically confuses me. For example, in the second one it says, "how many different ways are there to draw a line on this paper." If it had asked it even like, "How many different lines can be drawn." I would understand, but why are we not counting both different ways to draw the same line? Drawing a line from A to B seems like a different way than from B to A.