Two basic combinational methods questions

Question

I am an absolute beginner in math. I have 2 questions about combinatorial methods, with their answers. I spent lots of time to understand these questions and tried to find a connection how the author found this solution, but I am literally lost. Can anybody explain me very basically, how the author solved this questions?

Answer 1

So, I will answer the second question for you, and also I highly recommend you provide some sort of input as to whether you have learned basic problem-solving in Combinatorics.
Now, the question says that there are $4$ friends, say $A, B, C, D$ and there are $8$ distinct books to be given to them, such that the first $2, A$ and $B$ have $4$ books in total. So, first, we need to select $4$ out of the $8$ books to be given to the $2$ friends and there are $\binom 8 4$ ways of doing that.
Also then we have to give the books out to the $2$ people, and since we have $2$ options for each book, we have $2^4$ ways of giving the books to the first $2$ friends.
Next, we have to give the remaining $4$ to the other $2$ and this can also be done in $2^4$ ways since we can give each book to one of them. So finally in conclusion you get, $$\binom 8 4 \cdot 2^4 \cdot 2^4$$

Answer 2

Answer 1)
Total possible outcome=(total no.of options)^no. of times taken. So, it is $3^7$. No. of combination$=\binom nr$. So favourable outcomes for one student $=\binom72$. So for 2 students, it is $2^{7-2}$ chances. So the result is $\frac{\binom72 2^5}{3^7}$.