How many sets of all possible number pairs with each set having 6 numbers from 1-10 , what am I doing wrong?

Question

My Understanding

$n$= numbers to choose between $1$-$n$ i.e. 10

$r$ = match count i.e. 2

$q$ = count of numbers allowed to choose in 1 seti.e. 6

Note : While choosing a set, numbers cant be repeated and order doesn't matter. After each set all numbers are put back to choose next set.

possible combinations $r$ numbers at a time out of $n$ numbers $c_{total}= C(n,r)$

possible combinations $r$ numbers in 1 set (has $q$ numbers)$c_{set-total} = C(q,r)$

since each set already has $C(q,r)$
total number of sets $S =c_{total}/c_{total} = C(n,r)/C(q,r)$ Is this even correct ?

In fact sometime back I was able to logically conclude $S < c_{total}/c_{total} = C(n,r)/C(q,r)$ but now I am completely blacked out.

by using this for the said values $n=10, r=2, q=6$

then $c_{total}=45 ,c_{set-total}=15$

there for $S=3$

In $3$ sets I should get all possible pair of number between $1-10$

I stumbled upon La Jolla Covering Repository which says it is $4$ .

What am I doing wrong? Also how do I generate these sets with values, is there a logic?

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