Question

There are 5 doors to a lecture hall. The number of ways that a student can enter the hall and leave it by a different door is
A. 20
B. 16
C. 19
D. 21

Answer 1

We can calculate the number of ways of selecting a door to enter a lecture from 5 doors by using a combination, $^n{C_r}$, where $r$ objects are selected from $n$ objects. Then, find the number of ways of leaving the hall from a different door. At last, multiply both the number of ways to get the required answer.

Complete step by step solution:
There are 5 doors from where a student can enter.
And we know that the number of ways in which $r$ objects can be selected from $n$ objects is given by $^n{C_r}$ which is equal to $\dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
Here, 1 door has to be selected from 5 ways. Then, the number of ways of doing it is,
$\dfrac{{5!}}{{1!\left( {5 - 1} \right)!}} = \dfrac{{5!}}{{4!}} = 5$
Also, we are given that a person who entered from a door cannot leave from the same door.
So, after entering from 1 door, a student will be left with 4 doors from where the student can leave.
Then, the number of ways a student can leave from different door is,
$^4{C_1} = \dfrac{{4!}}{{1!\left( {4 - 1} \right)!}} = \dfrac{{4!}}{{3!}} = 4$
Therefore, the number of ways that a student can enter a hall and leave it by a different door is $5 \times 4 = 20$

Hence, option A is correct.

Note:
Here, we have the concept of combination. To find the total ways, we will take the product and not the sum as both are compulsory conditions. The formula of $^n{C_r}$ is $\dfrac{{n!}}{{r!\left( {n - r} \right)!}}$