actually we are arranging books but not two selves, that is why divide by 2.
It's not like that
5 books
You inserted 2 sticks between 5 books to make a partition of 3
Total elements = 5+2
Now you gave to only 2 students because whatever is left at last is gave to last students
Like 2 | 2 | 1
Total 7 elements ( 5 books + 2 sticks )
Distribute to 2 child
1st will get = 2 books
2nd will get = 2 books
And we didn't take care for what 3rd student will get
Because whatever is left after giving to 1st and 2nd student we will gave it to 3rd student
That's why
From 7 elements we choose only 2
$\binom{5+2}{2}$
Here all books are different that's why we multiply with 5!
If all books are identical than it's = $\binom{5+2}{2} X \frac{5!}{5!}$
" Sticks are only concept that we introduce to simply solve such questions "
"If in this question it's given that books selves are also different "
Than it has to be =
$\binom{5+2}{2} X 5! X 2!$
