Question

At a platform there are 3 gates numbered 1, 2 and 3 In how many ways can 100 people get inside the platform (Give that at a time only one person can get inside the platform through one gate)

• A. $\frac{102!}{2!}$
• B. $\frac{102!}{3!}$
• C. $\frac{102!}{4!}$
• D. $\frac{1020!}{5!}$

Answer 1

The correct option is A $\frac{102!}{2!}$

100 people can pass through the 3 gates in
${}^{100+(3-1)}{C}_{3-1}$ ways ....(formula (${}^{n+r-1}{C}_{r-1}\left)\right)$
$={}^{102}{C}_2$ ways.
Since all the 100 people are different, they can be arranged in 100! ways.
Therefore, the total number of possible ways will be
$=100!.{}^{102}{C}_2$
$=100!\left(\frac{102!}{2!.100!}\right)$
$=\frac{102!}{2!}$