Question

The number of ways in which candidates $A_1,A_2,...,A_{10}$ can be ranked if
$\left(i\right)$ $A_1$ and $A_2$ are next to each other
$\left(ii\right)$ $A_1$ is always above $A_2$ are

• A. $725760,1814400$
• B. $725760,1814401$
• C. $725760,1814402$
• D. $725761,1814400$

Answer 1

Answer: (A)

$725760,1814400$

$A_1$ and $A_2$ are always next to each other so lets take them as $1$ unit so, we have $9!$ ways of arranging them$A_1$ and $A_2$ can themselves arrange in $2!$ waysso there are $9!\times 2$ ways = $725760$ ways
$A_1$ is always on the top of $A_2$ so, we have $10!$ ways of arranging it,now for exactly half of the possibility, $A_1$ will be on the top of $A_2=1/2$, so there are $\frac{10!}{2}$ ways $=1814400$ ways
Hence, option 'A' is correct.