Question

There are 15 stations on a train route and the train has to be stopped at exactly 5 stations among these 15 stations. If it stops at at least two consecutive stations, then the number of ways in which the train can be stopped is

• A. $^{11}C_5$
• B. $^{15}C_5$
• C. $^{15}C_5 - ^{11}C_5$
• D. $^{15}C_{10} - ^9C_5$

Hint:

When a counting problem involves phrases like "at least one" or "at least two," it's often easier to count the total number of cases and subtract the number of cases where the condition is not met (the complement). The formula $^{n-k+1}C_k$ for selecting $k$ non-consecutive objects from $n$ is a very useful shortcut.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
This is a combinatorial problem that is best solved using the principle of complementary counting. The total number of ways to choose the stops is calculated, and then the number of ways for the opposite condition (no two stops are consecutive) is subtracted from the total.
Step 2: Key Formula or Approach
1. Total ways: The total number of ways to choose 5 stations out of 15 without any restrictions. This is given by the combination formula $^nC_k$.
2. Ways with no consecutive stops: The number of ways to choose 5 stations such that no two chosen stations are consecutive. This can be calculated using a standard combinatorial formula or the "gaps" method.
3. Required ways: The number of ways with at least two consecutive stops is (Total ways) - (Ways with no consecutive stops).
The formula for choosing $k$ non-consecutive items from $n$ items arranged in a line is $^{n-k+1}C_k$.
Step 3: Detailed Explanation
1. Total number of ways:
The total number of ways to choose 5 stations out of 15 is given by:
\[ \text{Total ways} = {^{15}}C_5 \] 2. Number of ways with no two consecutive stops:
We want to choose 5 stations out of 15 such that no two are consecutive. We use the formula $^{n-k+1}C_k$ with $n=15$ (total stations) and $k=5$ (stations to choose).
\[ \text{Ways with no consecutive stops} = {^{15-5+1}}C_5 = {^{11}}C_5 \] *Alternative (Gaps method):* Imagine the 10 stations where the train does not stop (N) arranged in a row: N N N N N N N N N N. These create 11 possible gaps (including the ends) where the 5 stopping stations (S) can be placed: \_ N \_ N \_ ... \_ N \_. To ensure no two S's are consecutive, we must place each of the 5 S's in a different gap. The number of ways to choose 5 distinct gaps from the 11 available is $^{11}C_5$.
3. Number of ways with at least two consecutive stops:
Using the principle of complementary counting:
\[ \text{Required ways} = (\text{Total ways}) - (\text{Ways with no consecutive stops}) \] \[ \text{Required ways} = {^{15}}C_5 - {^{11}}C_5 \] Step 4: Final Answer
The number of ways in which the train stops at at least two consecutive stations is $^{15}C_5 - ^{11}C_5$.