Let $S$ be the set of $6$ digits numbers $a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}$ (all digits distinct) where $a_{1}>a_{2}>a_{3}<a_{4}>a_{5}<a_{6}$. Then $n(S)$ is equal to

Which of the following is equal to $\dfrac{1.3.5....(2n-1)}{2.4.6....(2n)}$?

The letters of word $OUGHT$ are written in all possible orders and these words are written out as in a dictionary. Find the rank of the word $TOUGH$ in this dictionary.

In a conference hall 10  executive are to be seated on 10 chairs placed left to right. Executive $E_1$ wants to sit to the right of $E_2$ Also, $E_4$ wants to sit the left of $E_1$ and $E_5$ wants to sits to the left of ways $E_4$ Number of ways the executives can be seated on the 10 chairs, is