Question

There are two gates to enter a school and 3 staircases from the first floor to the second floor. How many possible ways are there for a student to go from outside the school to a classroom on the second floor and come back?

Answer 1

Hint: To go from outside the school to a classroom first select one gate from the given 2 gates by using the formula ${}^{2}{{C}_{1}}$. Now find the number of ways to select one staircase from the given 3 by using the expression ${}^{3}{{C}_{1}}$. Multiply these numbers to get the total number of ways to go to the classroom. Similarly, find the number of ways to go outside the school from this classroom. Take the product of the number of ways obtained to get the answer.

Complete step-by-step solution:
Here, we have been provided with the information that there are 2 gates to enter a school and 3 staircases from the first floor to the second floor. We have to find the total number of possible ways for a student to go to a classroom on the second floor and come back. Now, since there are 2 gates to enter the school, so the student can select any one of them. So, we have,
Number of ways to select 1 gate $={}^{2}{{C}_{1}}=2$
So, there are 2 ways to enter the school. Now, the student has to go to the second floor from the first floor by using any one of the 3 staircases available. So, the student has to select one staircase from the 3 staircases. So, we have,
Number of ways to select 1 staircase $={}^{3}{{C}_{1}}=3$
Therefore, the number of ways to go to the classroom from outside the school $=3\times 2=6$
Now, one may note that the student has to come back also. Since the number of staircases and gates are not changed, so the total number of ways of selection will remain the same. So, we have,
Number of ways to go outside the school from the classroom $=6$
Here, we can see that the student has to perform both the functions, that is go to the classroom and then return back, therefore the number of ways to perform both the functions will be the product of the number of ways to perform each function. Therefore, we have,
$\Rightarrow $ Number of ways to go to the classroom and return back $=6\times 6=36$
So, there can be 36 combinations possible. Hence, 36 is our answer.

Note: One may note that the student is performing both the functions and that is why we multiplied the number of ways of performing individual functions. If we would have been provided with the condition that the student has to either go to the classroom or return back then we would have considered the sum of the number of ways of performing individual functions. Remember the formula of combinations ${}^{n}{{C}_{r}}$ to solve the question.