In a $12$-storey house ten people enter a lift cabin. It is known that they will leave the lift in pre-decided groups of $2,3,$ and $5$ people at different storeys. The number of ways they can do so if the lift does not stop at the second storey is

The correct option is C

$720$

Explanation for the correct option.

Find the required number of ways.

Three groups of people enter the lift of a $12$ storey building in the ground floor and it is known that the lift does not stop on the second floor.

So lift will stop at three floors from the total of $10$ floors (excluding the ground and second floor).

So the number of ways three groups of people can leave the lift cabin is ${}^{10}P_3$.

$\begin{array}{rcl}{}^{10}P_3& =& \frac{10!}{(10-3)!}\left[{}^{n}P_r=\frac{n!}{(n-r)!}\right]\\ & =& \frac{10·9·8·7!}{7!}\\ & =& 720\end{array}$

So the required number of ways is $720$.

Hence, the correct option is C.