Calculate Number of Ways Persons Exit Lift at Floors

To find the number of ways in which $4$ persons exit at one floor and the remaining $5$ persons exit at a different floor, we follow these logical steps:

1. Determine the number of available floors for exiting: The building has a ground floor and $10$ higher floors ($1^{\text{st}}$ through $10^{\text{th}}$ floor). Since the lift does not stop at the $1^{\text{st}}$ and $2^{\text{nd}}$ floors, the eligible floors for exiting are: $\text{Floors available} = \{3^{\text{rd}}, 4^{\text{th}}, 5^{\text{th}}, 6^{\text{th}}, 7^{\text{th}}, 8^{\text{th}}, 9^{\text{th}}, 10^{\text{th}}\}$ Thus, the total number of available floors is $10 - 2 = 8$.

2. Form groups of persons: We need to select $4$ persons out of $9$ to exit at one floor, leaving the remaining $5$ persons to exit together at another floor. The number of ways to divide $9$ persons into two groups of $4$ and $5$ is given by: $\binom{9}{4} = \frac{9!}{4! \times 5!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126$

3. Assign floors to the two groups: Since the two group sizes are distinct ($4 \neq 5$), assigning floors involves choosing $2$ distinct floors out of the $8$ available floors in order (one for the 4-person group and one for the 5-person group): ${}^{8}P_{2} = 8 \times 7 = 56$

4. Calculate the total number of ways: Using the fundamental principle of counting, the total number of ways is: $\text{Total ways} = \binom{9}{4} \times {}^{8}P_{2} = 126 \times 56 = 7056$

Hence, the correct option is C.