In how many ways can $12$ balls be distributed between two friends such that one gets $8$ balls and the other gets $4$ balls?

$^nC_r + 2^nC_{r-1} + ^nC_{r-2} = $

$A$ bag contains $5$ red marbles,$4$ black marbles,and $3$ white marbles. The number of ways in which $4$ marbles can be drawn so that at most $2$ of them are red is:

$A$ question paper is divided into two parts,$A$ and $B$,each consisting of $5$ questions. If a candidate has to select $6$ questions in total,in how many ways can they do so if they must select at least $2$ questions from each part?

$A$ team of $5$ students is to be selected from $12$ students. If two particular students are to be included in that team,then the number of ways that such team can be selected is

There are two urns. Urn $A$ contains $3$ distinct red balls,and Urn $B$ contains $9$ blue balls. If two balls are selected at random from each urn and swapped,in how many ways can this operation be performed?