In how many ways can $5$ boys and $3$ girls sit in a row so that no two girls are together?

There are $6$ members to be selected for a group to be sent abroad from a total of $12$ members. In how many ways can the selection be made such that a particular member is always excluded?

The set $S = \{1, 2, 3, \ldots, 12\}$ is to be partitioned into three sets $A, B, C$ of equal size. Thus $A \cup B \cup C = S$ and $A \cap B = B \cap C = C \cap A = \emptyset$. The number of ways to partition $S$ is

$A$ box contains two white balls,three black balls,and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw?

In how many ways can a cricket team of $11$ players be selected out of $16$ players if $2$ particular players are to be included and $1$ particular player is to be rejected?

If $C(n, 8) = C(n, 6)$,find $C(n, 2)$.