# Simple probability proof

phospho
An enclyclopededia consisting of n (## n \geq 4 ##) similar volumes is kept on a shelf with the volumes in correct numerical order: that is, with volume 1o n the left, volume 2 next, and so on. The volumes are all taken down for cleaning and are replaced on the shelf in a random order. Prove that the probabilities of finding exactly n, (n-1), (n-2), (n-3) volumes in their correct positions on the shelf are, respectively, ## \dfrac{1}{n!} ##, ## 0 ##, ## \dfrac{1}{(n-2)!2} ##, ## \dfrac{1}{(n-3)!3} ##

I've done all of this, except I cannot construct a proof for (n-1) and 0

It seems very trivial, if one of the books is in the wrong place, then that means another book is in the wrong place, which means another book is in the wrong place and so on, therefore the probability of a book being in the correct position is 0, but how do I go about constructing an actual proof for it?

## Answers and Replies

Homework Helper
Gold Member
An enclyclopededia consisting of n (## n \geq 4 ##) similar volumes is kept on a shelf with the volumes in correct numerical order: that is, with volume 1o n the left, volume 2 next, and so on. The volumes are all taken down for cleaning and are replaced on the shelf in a random order. Prove that the probabilities of finding exactly n, (n-1), (n-2), (n-3) volumes in their correct positions on the shelf are, respectively, ## \dfrac{1}{n!} ##, ## 0 ##, ## \dfrac{1}{(n-2)!2} ##, ## \dfrac{1}{(n-3)!3} ##

I've done all of this, except I cannot construct a proof for (n-1) and 0

It seems very trivial, if one of the books is in the wrong place, then that means another book is in the wrong place,

Yes.

which means another book is in the wrong place and so on,

Why? Couldn't just two books be swapped?

phospho
Yes.

Why? Couldn't just two books be swapped?

I guess, but that still leaves the probability to be 0, as only 1 book being in the wrong place means at least two are in the incorrect position

Homework Helper
Gold Member
I guess, but that still leaves the probability to be 0, as only 1 book being in the wrong place means at least two are in the incorrect position

That's correct. You can't have just one in the wrong place, and that's why that probability is zero.

phospho
Is that a proof?