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## Homework Statement

"Let i be a fixed interger where 1≤i≤n-2. Given the graph K

_{n,n}write down a formula for the number of ways you can choose a pair of distinct 1-factors which intersect in precisely i edges? Hint: fix one of the 1-factors, choose the i edges which will be common to both and then use derangements to choose the other n-i edges. Now how many ways can you choose the first 1-factor and when are you doubling counting?

## Homework Equations

!n = n!

^{n}Ʃ

_{r=0}(-1)

^{r}/r!

## The Attempt at a Solution

Following the hint, should I approach it by setting a 1 factor of A = {e

_{1}, e

_{2}, e

_{3}... e

_{n}} And then set the intersection of A and the other 1-factor B as A[itex]\cap[/itex]B = {e

_{1}, e

_{2}..... e

_{i}}

Problem is I don't know how to relate that idea to the formula we've been using - like what values to use for n and r? I would think n=(n-i), but that doesn't seem right. Am I completely off? Really having trouble with this one. :/ Thanks for any help!