# Thread: How many functions [5]--->[5] have at least one fixed point?

1. ## How many functions [5]--->[5] have at least one fixed point?

If f is a function and f(i) = i then we call i a fixed point of f.

a) How many functions [5] $\rightarrow$[5] have at least one fixed point?

I have been working on this for hours! I've written out [3] $\rightarrow$[3], and [4] $\rightarrow$[4].

$[\sum_{k=0}^{n-3}(n-k)(n-2)]+1$

2. Originally Posted by oldguynewstudent
If f is a function and f(i) = i then we call i a fixed point of f.

a) How many functions [5] $\rightarrow$[5] have at least one fixed point?

I have been working on this for hours! I've written out [3] $\rightarrow$[3], and [4] $\rightarrow$[4].

$[\sum_{k=0}^{n-3}(n-k)(n-2)]+1$
How about thinking of it this way: what you want is the total number of functions minus the ones with NO fixed points.

How far can you get with that conceptualization?

3. Originally Posted by Jhevon
How about thinking of it this way: what you want is the total number of functions minus the ones with NO fixed points.

How far can you get with that conceptualization?
I have been working it from that angle also. I came out with $4^5$ with no fixed points, but I could not justify that either.

4. Originally Posted by oldguynewstudent
I have been working it from that angle also. I came out with $4^5$ with no fixed points, but I could not justify that either.
yes, you have 5 elements, each of which has 4 choices of what to map to, that is, $4^5$ functions. So the overall answer would be...?

5. Originally Posted by Jhevon
yes, you have 5 elements, each of which has 4 choices of what to map to, that is, $4^5$ functions. So the overall answer would be...?
Yes thanks! That is the original answer I came up with 2 or 3 hours ago but then I thought it was wrong! $5^5 - 4^5$

6. Originally Posted by oldguynewstudent
Yes thanks! That is the original answer I came up with 2 or 3 hours ago but then I thought it was wrong! $5^5 - 4^5$
Yup, that's it