There are intermediate railway stations - RRB Mathematics

Selection of Non-Consecutive Intermediate Railway Stations

A train travels from one terminal to another, passing through $P$ intermediate railway stations. The problem is to determine the number of ways the train can stop at exactly three of these intermediate stations, with the restriction that no two selected stations are consecutive. This scenario involves combinatorial selection under a non-consecutiveness constraint.

Problem Statement and Constraints

The task is to select three intermediate stations from a total of $P$ such that no two chosen stations are consecutive. The selection must be made from the intermediate stations only, not including the terminals…

Selection of Non-Consecutive Intermediate Railway Stations

A train travels from one terminal to another, passing through $P$ intermediate railway stations. The problem is to determine the number of ways the train can stop at exactly three of these intermediate stations, with the restriction that no two selected stations are consecutive. This scenario involves combinatorial selection under a non-consecutiveness constraint.

Problem Statement and Constraints

The task is to select three intermediate stations from a total of $P$ such that no two chosen stations are consecutive. The selection must be made from the intermediate stations only, not including the terminals at either end.

• Total intermediate stations: $P$
• Stations to be selected: $3$
• Restriction: No two selected stations are consecutive

Representation of Stations

The stations can be represented as a sequence, with the first and last being the terminals and the $P$ stations in between labeled as $S_1, S_2, ..., S_P$. The selection is to be made from $S_1$ to $S_P$.

• Terminals: Not included in selection
• Intermediate stations: $S_1, S_2, ..., S_P$

Combinatorial Principle for Non-Consecutive Selection

When selecting $r$ items from $n$ such that no two are consecutive, a specific combinatorial formula is used. The transcript states the formula for such cases, which is essential for solving the problem.

• Formula for non-consecutive selection: If $r$ items are to be chosen from $n$ such that no two are consecutive, the number of ways is $n - r + 1 \choose r$.

Application to the Given Problem

Applying the above principle to the current scenario, where $r = 3$ and $n = P$, the formula becomes:

• Number of ways: P - 3 + 1 \choose 3 = P - 2 \choose 3

Stepwise Reasoning

The reasoning behind the formula is based on ensuring that after selecting a station, at least one unselected station separates it from the next selected station. This reduces the effective number of available positions.

• After each selection: At least one station must remain unselected between any two selected stations.
• Effective positions: The formula accounts for the required gaps by reducing the total number of available positions.

Final Formula and Result

The final answer for the number of ways to select three non-consecutive intermediate stations from $P$ is:

Key Takeaways

Selecting items with a non-consecutiveness constraint reduces the effective pool of choices. The formula $n - r + 1 \choose r$ is directly applicable when no two selected items can be consecutive, as explicitly stated in the transcript.

• Direct application: Use $n - r + 1 \choose r$ for non-consecutive selection.
• For this problem: Substitute $n = P$, $r = 3$ to get $P - 2 \choose 3$.

Common Pitfall

A common mistake is to use the standard combination formula $n \choose r$ without accounting for the non-consecutiveness restriction. This leads to an overcount of cases where selected stations are consecutive, which is not allowed.

• Incorrect approach: Using $P \choose 3$ ignores the restriction.
• Correct approach: Use $P - 2 \choose 3$ as derived.

Answer Identification

The correct answer, as per the transcript, is option B, which corresponds to $P - 2 \choose 3$. This is the only answer that satisfies the non-consecutiveness condition for selecting three intermediate stations.