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.