Seven students are to be allotted seven beds in a hostel hall. Two of them, Parvin and Anju, refuse to take adjacent beds because Anju snores. In how many different ways can all seven students be allotted the seven beds so that Parvin's bed is not next to Anju's bed?

Introduction / Context:
This arrangement problem adds a restriction of non adjacency for two specific students. We must count permutations of seven distinct students with the condition that Parvin and Anju are not seated or allotted beds next to each other. It demonstrates the use of complementary counting in permutations.

Given Data / Assumptions:

• There are 7 distinct students.
• There are 7 distinct beds in a line.
• Parvin and Anju must not occupy adjacent beds.
• Any allotment is a one to one assignment of students to beds.

Concept / Approach:
Instead of directly counting the valid arrangements, it is easier to use the complement. First count all possible arrangements of the 7 students without restrictions. Then subtract arrangements in which Parvin and Anju are adjacent. The remaining arrangements are exactly those where Parvin and Anju are not next to each other.

Step-by-Step Solution:
Step 1: Total unrestricted arrangements of 7 distinct students in 7 distinct beds is 7! = 5040.Step 2: To count arrangements where Parvin and Anju are adjacent, treat them as a single block PA or AP.Step 3: The block (PA or AP) plus the other 5 students makes 6 entities to arrange.Step 4: The number of ways to arrange these 6 entities in a line is 6! = 720.Step 5: Inside the block, Parvin and Anju can switch places in 2 ways: PA or AP.Step 6: Therefore, total arrangements where they are adjacent = 6! * 2 = 720 * 2 = 1440.Step 7: Now subtract these from the total: valid arrangements = 5040 - 1440 = 3600.

Verification / Alternative check:

Why Other Options Are Wrong:

• 2400 and 6400: These do not align with 7! minus a reasonable multiple of 6! and likely result from miscounting the adjacency cases or forgetting the internal arrangements of the block.
• 7200: This is larger than the total number of possible arrangements (5040) and so cannot be correct.

Common Pitfalls:
Some learners count the adjacent cases incorrectly by treating the block as a single entity but forgetting to multiply by 2 for the internal arrangement of Parvin and Anju. Others mistakenly use 5! instead of 6! for the outside arrangements. Another frequent error is to forget to subtract from the total and directly use the count of adjacent cases as the answer.

Final Answer:
The number of ways to allot the beds so that Parvin and Anju are not next to each other is 3600.