If $^{n^2 - n}C_2 = ^{n^2 - n}C_{10}$,then $n = $

The number of shortest paths from point $A$ to $D$ (as shown in the figure) is:

There are $6$ men and $8$ women in a club. $A$ committee of $5$ people has to be formed. What will be the number of ways to select $5$ people if $2$ men and $3$ women are selected?

How many even numbers of four digits can be formed with the digits $0, 1, 2, 3, 4, 5$ and $6$,if no digit is used more than once?

How many $3$-letter words can be formed from the letters of the word '$EXPERTISE$',if repetition of letters is not allowed?

Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated?