If $^nC_{12} = ^nC_6$,then $^nC_2 = $

Number of shortest paths from $HOSTEL$ to $ALLEN$ is equal to (as shown in the given figure):

$A$ parallelogram is cut by two sets of $m$ lines parallel to its sides. The number of parallelograms thus formed is

Consider a class of $5$ girls and $7$ boys. The number of different teams consisting of $2$ girls and $3$ boys that can be formed from this class,if there are two specific boys $A$ and $B$ who refuse to be members of the same team,is

In how many different ways can the letters of the word '$LEADING$' be arranged in such a way that the vowels always come together?

How many different ways can the letters in the word $ATTEND$ be arranged?