Question

In a train five seats are vacant, then how many ways can three passengers sit?
1. 20
2. 30
3. 10
4. 60

Answer 1

Hint: For solving this question you should know about the permutation and combinations. As we know that this problem has statements only and we have to tell whether these are true or false. So, first we will discuss the permutation and combination and then by applying the concept of these, we will find the solution of the problem and then we will find the existence of the given statements.

Complete step-by-step solution:
According to our question, it is given that in a train there are 5 seats vacant. And three passengers have to sit on them. So, we have to determine the number of ways in which all the three passengers can sit on them. We will use combinations here because by that we can calculate it easily.
As we know, permutations are ordered combinations. It means where the order does not matter, there it is a permutation. And permutations have also two types, one is repeated or repetition is allowed and the second is no repetition. The permutations with repetition are the easiest to calculate. And the permutation with no repetition reduces the number of available choices each time. So, if we see our question, then:
Total number of seats available in the train = 5
Total number of passengers for sitting on them = 3
So, here,
The number of ways three passengers can sit on the five seats = ${}^{5}{{P}_{3}}$.
$=\dfrac{5!}{\left( 5-3 \right)!}=\dfrac{5!}{2!}=\dfrac{120}{2}=60$
Hence the correct answer is option 4.

Note: While solving this type of questions you have to ensure that we will use the permutation or combination here and, in the permutation, it is necessary to find that if there are any events that are repeating or not repeating.