Question

Three persons entered a railway compartment in which 5 seats were vacant. Find the number of ways in which they can be seated
A) 30
B) 45
C) 120
D) 60

Answer 1

Hint: We have been given the number of persons and the vacant seats where they can be seated. We can find the number of ways for doing so by using the concept of permutation and combination.
The formula for combination is:
 $ ^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}} $ where n are the total number of things and n are the things to be selected.
For the calculation of factorial, we take the product of numbers from that number to 1

Complete step-by-step answer:
As we have to find the numbers of ways in which three persons can be seated in 5 vacant seats, we can use the concept of permutation and combination.
Permutation is for arranging the members of a set into a sequence while combination is the way of selecting items from a group where the order does not matter.
When r things are selected from n things, then the combination formula is given as:
 $ ^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}} $
According to the given statement, we have to find the number of ways in which 3 persons are to be arranged in 5 vacant seats.
These persons can select any seat in any order, so there way of selection is given by combination as:
 $ {\Rightarrow ^5}{C_3}\_\_\_(1) $
But these 3 people can also change the order among themselves whose number ways will be given by $ n! $ where n will be 3.
 $ \Rightarrow 3!\_\_\_(2) $
So, the total number of ways (N) are given by the product of (1) and (2) as:
 $ N{ = ^5}{C_3} \times 3! $
Applying the formula of combination and calculating the factorial by the multiplication from the number to 1.
 $
  \Rightarrow N = \dfrac{{5!}}{{3! \times 2!}} \times 3! \\
  \Rightarrow N = \dfrac{{5!}}{{2!}} \\
  \Rightarrow N = \dfrac{{5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1}} \\
   \Rightarrow N = 5 \times 4 \times 3 \\
   \Rightarrow N = 60 \\
  $
Therefore, the number of ways in which three persons can be seated if 5 seats were vacant is 60 and the correct option is D).
So, the correct answer is “Option D”.

Note: This question can also be solved by fixing and eliminating of positions as:
We have 5 seats and 3 persons to be seated.
The first person can choose any of the 5 seats
 1P = 5 ways
When one seat is fixed, the second person has 1 seat less to choose from
2P = (5-1) ways
2P = 4 ways
For, the third person, two seats are fixed already and he now has two seats less to choose from