Question

There are 12 intermediate stations on a railway line from one terminus to another. Number of ways in which a train can stop at 5 of these intermediate stations, if no two of these stopping are to be consecutive
$
  {\text{A}}{\text{. 56}} \\
  {\text{B}}{\text{. 495}} \\
  {\text{C}}{\text{. 792}} \\
  {\text{D}}{\text{. 70}} \\
$

Answer 1

Hint:-In this question first we examine the number of non-stopping intermediate stations among 12 intermediate stations. Since, no two stopping stations are to be consecutive so we can place them in between non-stopping stations and calculate it.

Complete step by step answer:
Since, there are 5 intermediate stations where trains can stop out of 12 intermediate stations on a railway line from one terminus to another.
 Then, we left with the 12-5 =7 non-stopping stations on a railway line.
Now, there are 8 positions to place 5 stopping stations such that no two stopping stations are consecutive in between these 7 non-stopping stations. Since, on both sides of non-stopping stations stopping stations can be placed.
This can be done in ${}^8{C_5}$ ways.
  $\Rightarrow {\text{Number of ways for placing 5 stopping stations }} = _{}^8{C_5}$
  ${\text{ }} = \dfrac{{8!}}{{3! \times 5!}}$
  ${\text{ }} = 56 $

Hence option A. is correct.

Note:- Whenever you get this type of problem the key concept to solve is to visualize the situation and then use permutation and combination to get the required result. This problem can also be solvable by the concept of Multinomial Expansion in Combination but in that way it is a bit lengthy.