MHB Calculating Probability for Non-consecutive Lockers in a Discrete Model

[Abstract3000]

Hello,
I have a question I am trying to figure out how it works and I am so confused I need a break down of what is exactly going on with this problem

the Question.
"Concern three persons who each randomly choose a locker among 12 consecutive lockers"

What is the probability that no two lockers are consecutive?

The answer given that confuses me even more:
-----------------------------------------------------------------------
1st to find ways exactly 2 lockers are consecutive

X X _ _ _ _ _ _ _ _ _ _ 9 Ways
_ X X _ _ _ _ _ _ _ _ _ 8 Ways
_ _ X X _ _ _ _ _ _ _ _ 8 Ways

5 others w/8

_ _ _ _ _ _ _ _ X X _ _ 8
_ _ _ _ _ _ _ _ _ X X _ 8
_ _ _ _ _ _ _ _ _ _ X X 9

# = 9X2 + 9X8 = 90
Next # Ways w/3 consecutive = 10 start in 1,2,...,10

== P(no 2 consec) = 1 - 100/C(12,3) ~ .545 = 5.45X10^10-1
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This is the teachers solution and I have absolutely no idea on how he got all the number he did I am really lost (on all of it, the solution makes no sense in what he has written down I don't get where he gets the 8's the 9's or the 10 from), anyone understand this and have the time to break it down for me?

Thanks!

 

[Plato2]

Re: Discrete Probability Quetion

"Concern three persons who each randomly choose a locker among 12 consecutive lockers"
What is the probability that no two lockers are consecutive?

This is the teachers solution and I have absolutely no idea on how he got all the number he did I am really lost (on all of it, the solution makes no sense in what he has written down I don't get where he gets the 8's the 9's or the 10 from), anyone understand this and have the time to break it down for me?

I do not follow that solution either. But here is a model.
Think of a string 111000000000, the 1's represent the chosen lockers and the 0's empty.
100100000100 in that model no two chosen lockers are consecutive.
But in 001000001100 in that model two chosen lockers are consecutive.

So how many ways can we rearrange the string 111000000000 so no two 1's are consecutive?

\_\_0\_\_0\_\_0\_\_0\_\_0\_\_0\_\_0\_\_0\_\_0\_\_ Note that the nine 0's create ten places that we can place the 1's so no two are consecutive.

Here is the caculation.