I have trouble understanding the 8/9 step when they switch the denominators after converting the division to a multiplication, the step after the equal sign (when the 1! and 7! becomes 7! and 1!) also in the step 9/9 why there is a factorial of 3 posibilities (7* 6* 5) now instead of 4 possibilities (7*6*5*4) like in the main question?
(couldn't add a proper image markup due to no reputation, but this questions has no point without the image)
We can do this question in a simple way.
Choose 1st child. $P(boy)=\frac{5}{7}$ because 5 boys and 2 girls.
Now choose 2nd child. $P(boy)=\frac{4}{6}$ because 4 boys and 2 girls.
Similarly, third and fourth student. We will get $\frac{3}{5}$ and $\frac{2}{4}$ respectively.
$\frac{5}{7}*\frac{4}{6}*\frac{3}{5}*\frac{2}{4}=\frac{1}{7}$.