If $a, b$ and $c$ are the greatest value of $^{19} \mathrm{C}_{\mathrm{p}},^{20} \mathrm{C}_{\mathrm{q}}$ and $^{21 }\mathrm{C}_{\mathrm{r}}$ respectively, then

In a conference of $8$ persons, if each person shake hand with the other one only, then the total number of shake hands shall be

The number of ways in which thirty five apples can be distributed among $3$ boys so that each can have any number of apples, is

Consider $4$ boxes, where each box contains $3$ red balls and $2$ blue balls. Assume that all $20$ balls are distinct. In how many different ways can $10$ balls be chosen from these $4$ boxes so that from each box at least one red ball and one blue ball are chosen?

Determine $n$ if

$^{2 n} C_{3}:^{n} C_{3}=11: 1$