$$\frac{7 !-6 !+5 ! \cdot 4 !}{6 !}$$ is the fraction I want to evaluate. Dealing with this, I come up with the following wrong expansion. What point do I forget?
$$ = \frac{4! \cdot{\color{Green} (}(5 \cdot 6 \cdot 7-5 \cdot 6)+5\color{Green} )}{6!} $$
$$= \frac{4! \cdot((210-30)+5)}{6!}$$
$$ = \frac{4! \cdot185}{6!} = \frac{185}{5\cdot6}$$
It should be
$$\frac{4!(5\cdot6\cdot7-5\cdot6+5!)}{6!}$$
This is since you have the term $5!\cdot4!$ where you can factor out a $4!$ term.
$$=\frac{4!(5\cdot6\cdot7-5\cdot6+5!)}{6!}$$ $$=\frac{4!(180 + 120)}{6!}$$ $$=\frac{24(300)}{720}$$ $$=\frac{10(720)}{720}$$ $$=10$$