Number of One to One functions such that $f(i)\neq i$ Given the sets $A=\{1,\dots,4\}$ and $B=\{1,\dots,7\}$ how many one to one functions are there from $A$ to $B$ such that $f(i)\neq i$ ?
I used inclusion exclusion to first find the number of one to one functions:
$S_1 = f(a)=f(b) , S_2 = f(a)=f(b)=f(c) , S_3 = f(a)=f(b)=f(c)=f(d)$
$$7^4 - \binom{4}{2}\cdot 7^3 + \binom{4}{3}\cdot 7^2 - \binom{4}{4}\cdot 7^1=532$$
I then repeated using a similar approach for eliminating cases of $f(a)=a$, etc...
$$532 - \binom{4}{1}\cdot6^3 + \binom{4}{2}\cdot5^2 - \binom{4}{3}\cdot4^1 + \binom{4}{4}\cdot 3^0$$
Are my solutions correct?
EDIT
I see that the number of one to one functions is $7\cdot6\cdot5\cdot4=840$ and that makes sense.  But I still can't seem to get it to work using inclusion exclusion.
$U$ would be the group of all possible functions, $7^4=2401$.
$S_1 = $ where $f(a)=f(b)$
$S_2 = $ where $f(a)=f(b)$ and $f(c)=f(d)$
$S_3 = $ where $f(a)=f(b)=f(c)$
$S_4 = $ where $f(a)=f(b)=f(c)=f(d)$
 A: Here's how it goes, using inclusion-exclusion to count the one-one functions. 
Notice that you have 6 conditions: $f(a)=f(b)$, $f(a)=f(c)$, $f(a)=f(d)$, $f(b)=f(c)$, $f(b)=f(d)$, and $f(c)=f(d)$. 
So the first three terms in the inclusion-exclusion are $$7^4-6\times7^3+{6\choose2}7^2$$ Now the next term is trickier. If the three conditions that hold are $f(a)=f(b)$, $f(a)=f(c)$, and $f(b)=f(c)$, that contributes $7^2$, but if the three conditions are $f(a)=f(b)$, $f(b)=f(c)$, and $f(c)=f(d)$, then the contribution is only $7$. So so far we have $$7^4-6\times7^3+{6\choose2}7^2-4\times7^2-16\times7$$ Now if 4, 5, or 6 conditions hold, the contribution is 7, so you get $$7^4-6\times7^3+{6\choose2}7^2-4\times7^2-16\times7+{6\choose4}7-6\times7+7$$ With any luck, I haven't missed anything, and that comes out to $840$ (and you can see that I wasn't kidding when I said this was the hard way). 
A: $$ f(1) \in \{2,3,4,5,6,7\} $$
$$ f(2) \in \{1,3,4,5,6,7\} $$
$$ f(3) \in \{1,2,4,5,6,7\} $$
$$ f(4) \in \{1,2,3,5,6,7\} $$
The total possible functions is:
$$ 6^4 = 1296 $$
