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There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. If N is the number of ways in which guests can be accommodated. Then the value of N6 is
Correct answer is '8'. Can you explain this answer?
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There are eight rooms on the first floor of a hotel, with four rooms ...
Clearly guests will stay either in '✓' or in '×'
Therefore, number of required ways =2 × 4! = 48
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There are eight rooms on the first floor of a hotel, with four rooms ...
Problem:
There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated. Four guests have to be accommodated in four of the eight rooms such that no two guests are in adjacent rooms or in opposite rooms. We need to find the value of N6.
Solution:
To solve this problem, we will use a combination of counting techniques and symmetry.
Step 1: Counting the total number of ways to accommodate the guests without any constraints
There are 8 rooms available for the guests to choose from. Since each guest can be accommodated in any of the 8 rooms, the total number of ways to accommodate the guests without any constraints is 8 * 7 * 6 * 5 = 1680.
Step 2: Considering the constraint of guests not being in adjacent rooms
Since there are 4 rooms on each side of the corridor, the guests can be accommodated in the following ways:
- 1 guest on each side, 2 guests on the other side: 2 * 4 * 3 * 2 = 48 ways
- 2 guests on one side, 1 guest on the other side: 4 * 3 * 2 * 4 = 96 ways
The total number of ways to accommodate the guests without being in adjacent rooms is 48 + 96 = 144.
Step 3: Considering the constraint of guests not being in opposite rooms
If a guest is placed in a room, then the opposite room cannot be occupied by another guest. Let's consider the following cases:
Case 1: 1 guest on each side, 2 guests on the other side
- Guest 1 can be accommodated in any of the 4 rooms on one side: 4 ways
- Guest 2 can be accommodated in any of the 3 remaining rooms on the same side: 3 ways
- Guest 3 can be accommodated in any of the 2 remaining rooms on the opposite side: 2 ways
- Guest 4 can be accommodated in the only remaining room on the opposite side: 1 way
The total number of ways in this case is 4 * 3 * 2 * 1 = 24.
Case 2: 2 guests on one side, 1 guest on the other side
- The 2 guests on one side can be selected in 4C2 = 6 ways
- The remaining 2 guests can be accommodated in the 2 remaining rooms on the other side in 2 ways
The total number of ways in this case is 6 * 2 = 12.
Step 4: Calculating the final answer
To find the total number of ways to accommodate the guests without any constraints, we calculated 1680 in Step 1. However, we want to count the number of ways that satisfy the given constraints. From Step 2, we know that there are 144 ways to accommodate the guests without being in adjacent rooms. From Step 3, we know that there are 24 ways in Case 1 and 12 ways in Case 2 to accommodate the guests without being in opposite rooms.
Therefore, the value of N6 (number of ways to accommodate the guests satisfying the given
There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated. Four guests have to be accommodated in four of the eight rooms such that no two guests are in adjacent rooms or in opposite rooms. We need to find the value of N6.
Solution:
To solve this problem, we will use a combination of counting techniques and symmetry.
Step 1: Counting the total number of ways to accommodate the guests without any constraints
There are 8 rooms available for the guests to choose from. Since each guest can be accommodated in any of the 8 rooms, the total number of ways to accommodate the guests without any constraints is 8 * 7 * 6 * 5 = 1680.
Step 2: Considering the constraint of guests not being in adjacent rooms
Since there are 4 rooms on each side of the corridor, the guests can be accommodated in the following ways:
- 1 guest on each side, 2 guests on the other side: 2 * 4 * 3 * 2 = 48 ways
- 2 guests on one side, 1 guest on the other side: 4 * 3 * 2 * 4 = 96 ways
The total number of ways to accommodate the guests without being in adjacent rooms is 48 + 96 = 144.
Step 3: Considering the constraint of guests not being in opposite rooms
If a guest is placed in a room, then the opposite room cannot be occupied by another guest. Let's consider the following cases:
Case 1: 1 guest on each side, 2 guests on the other side
- Guest 1 can be accommodated in any of the 4 rooms on one side: 4 ways
- Guest 2 can be accommodated in any of the 3 remaining rooms on the same side: 3 ways
- Guest 3 can be accommodated in any of the 2 remaining rooms on the opposite side: 2 ways
- Guest 4 can be accommodated in the only remaining room on the opposite side: 1 way
The total number of ways in this case is 4 * 3 * 2 * 1 = 24.
Case 2: 2 guests on one side, 1 guest on the other side
- The 2 guests on one side can be selected in 4C2 = 6 ways
- The remaining 2 guests can be accommodated in the 2 remaining rooms on the other side in 2 ways
The total number of ways in this case is 6 * 2 = 12.
Step 4: Calculating the final answer
To find the total number of ways to accommodate the guests without any constraints, we calculated 1680 in Step 1. However, we want to count the number of ways that satisfy the given constraints. From Step 2, we know that there are 144 ways to accommodate the guests without being in adjacent rooms. From Step 3, we know that there are 24 ways in Case 1 and 12 ways in Case 2 to accommodate the guests without being in opposite rooms.
Therefore, the value of N6 (number of ways to accommodate the guests satisfying the given
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There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. If N is the number of ways in which guests can be accommodated. Then the value of N6 isCorrect answer is '8'. Can you explain this answer?
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There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. If N is the number of ways in which guests can be accommodated. Then the value of N6 isCorrect answer is '8'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. If N is the number of ways in which guests can be accommodated. Then the value of N6 isCorrect answer is '8'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. If N is the number of ways in which guests can be accommodated. Then the value of N6 isCorrect answer is '8'. Can you explain this answer?.
There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. If N is the number of ways in which guests can be accommodated. Then the value of N6 isCorrect answer is '8'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. If N is the number of ways in which guests can be accommodated. Then the value of N6 isCorrect answer is '8'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. If N is the number of ways in which guests can be accommodated. Then the value of N6 isCorrect answer is '8'. Can you explain this answer?.
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