The above three questions are based on a single logic which is Indistinguishable Objects and Distinguishable Boxes (IODB) why??
Here positive integers don’t matter the same as objects but in which box you are putting that matters for example here there are two boxes in question (a) so both are distinguishable as 1 + 2 and 2 + 1 are different things.
To solve IODB problems think of it as a star-bar problem means objects are stars and the divider between two objects is a bar like in 1 + 2 → 1 and 2 are objects while + is considered as a bar.
Formula used in this IODB template is $\binom{total number of stars + total number of bars}{total number of stars}$ = $\binom{total number of stars + total number of bars}{total number of bars}$
Question a). There are n objects such that value of n >= 2 so at the beginning we will give 2 objects so now we will have n -2 objects or n -2 starts; since we want it as the sum of two positive integers so number of bar = 1.
total number of ways here will be $\binom{n-2 +1}{1}$ { It is simple to use as it is one of the templates in Combinatorics}
which is equal to $\binom{n-1}{1}$.
Question b). here there are n –3 stars and two bars so the answer will be $\binom{n-3 +2}{2}$ = $\binom{n-1}{2}$.
Question c). here we are already providing k objects so there are left with n – k objects so there are n – k stars and k bars
$\binom{n-k+k -1}{k -1}$ = $\binom{n – 1}{k – 1}$
{ PS: TO understand IODB template watch Goclasses combinatorics lecture I am not advertising that but that is really awesome. }