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 

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.

A password for a computer system requires exactly $6$ characters. Each character can be either one of the $26$ letters from A to Z or one of the ten digits from $0$ to $9$. The first character must be a letter and the last character must be a digit. How many different possible passwords are there?

Let $\displaystyle \mathrm{S}_{1}=\sum_{\mathrm{j}=1}^{10}j(j-1)^{10}\mathrm{C}_{\mathrm{j}}$ , $\displaystyle \mathrm{S}_{2}=\sum_{\mathrm{j}=1}^{10}j^{10}\mathrm{C}_{\mathrm{j}}$ and $\displaystyle \mathrm{S}_{3}=\sum_{\mathrm{j}=1}^{10}j^{2}10_{\mathrm{C}_{\mathrm{j}}}$.
Statement-1: $\mathrm{S}_{3}=55\times 2^{9}$
Statement-2: $\mathrm{S}_{1}=90\times 2^{8}$ and $\mathrm{S}_{2}=10\times 2^{8}$.