Let $A_1,A_2, \dots, A_m$ be $k$-subsets (each of size $k$) of $[n] := \{1, 2, 3, \dots, n\}$ such that for every pair $i, j \, (1\le i < j \le n)$ there exists some $A_l \, (1\le l \le m)$ such that $\{i, j\} \in A_l$.
What is the minimum value of $m$ and how do we enumerate/generate a family of subsets $A_1, A_2, A_3 , \dots, A_m$?
(This is not 2916733)
Let $m_{n,k} $ be the number you are looking for. There is a recursive relation between $m_{n,k} $ and $ m_{(n-1),k}$. Write $n-1=q(k-1)+r $ where $ 0\leq r<k-1$. Note that when you add a new element $a_{n} $ to the set $B_{n-1}:=\{a_1,\cdots,a_{n-1}\} $ there are $n-1 $ sets $\{i,j\}\subseteq B_{n-1} $ containing $a_n $ so you have to "add those pairs" to your old sets $A_1,\cdots, A_{m_{n,k}} $. The minimum way of doing this is by adding the $ k-$sets $\{a_{n+1},a_1,\cdots,a_{k-1}\}, \cdots, \{a_{n+1},a_1,\cdots,a_{qk-q}\}$ (there are $ q$ of them ) to your collection. Now you have to add a set cointaining all the remining $ r$ pairs (this is only one set because $ r<k-1$). So $ m_{(n-1),k}+q+1=m_{n,k}$.
Now construct an especific example of the situation and use the previous relation to make a conjecture of the value of $ m_{n,k}$.