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There are 8 seats in row, where 4 persons are to be seated. Then the probability of making them sit so that each person has exactly one neighbour, is
- a)1/14
- b)3/14
- c)1/7
- d)3/7
Correct answer is option 'C'. Can you explain this answer?
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There are 8seats in row, where 4persons are to be seated. Then the pro...
Problem:
There are 8 seats in a row, and we need to seat 4 people. The condition is that each person must have exactly one neighbor. We need to find the probability of arranging the people satisfying this condition.
Solution:
Step 1: Total number of seating arrangements:
To find the total number of seating arrangements, we need to consider that each person can occupy any of the 8 seats. Therefore, the total number of arrangements is given by:
Total arrangements = 8P4 = 8! / (8-4)! = 8! / 4! = 8 * 7 * 6 * 5 = 1680
Step 2: Counting favorable arrangements:
To count the number of favorable arrangements where each person has exactly one neighbor, we can break it down into cases.
Case 1: When the four people sit together in a block.
In this case, the 4 people can form a block and sit together in any of the 5 available spaces. Once the block is formed, the 4 people can arrange themselves within the block in 4! = 24 ways.
Total favorable arrangements in this case = 5 * 24 = 120
Case 2: When the four people sit with one empty seat between each person.
In this case, we can have 4 possible configurations where one person sits in the odd-numbered seats and the other person sits in the even-numbered seats. Each configuration can be arranged in 4! = 24 ways.
Total favorable arrangements in this case = 4 * 24 = 96
Step 3: Calculate the probability:
The probability of the favorable arrangements is given by the ratio of the number of favorable arrangements to the total number of arrangements.
Probability = (Total favorable arrangements) / (Total arrangements)
Probability = (120 + 96) / 1680
Probability = 216 / 1680
Probability = 1 / 7
Therefore, the correct answer is option C) 1/7.
There are 8 seats in a row, and we need to seat 4 people. The condition is that each person must have exactly one neighbor. We need to find the probability of arranging the people satisfying this condition.
Solution:
Step 1: Total number of seating arrangements:
To find the total number of seating arrangements, we need to consider that each person can occupy any of the 8 seats. Therefore, the total number of arrangements is given by:
Total arrangements = 8P4 = 8! / (8-4)! = 8! / 4! = 8 * 7 * 6 * 5 = 1680
Step 2: Counting favorable arrangements:
To count the number of favorable arrangements where each person has exactly one neighbor, we can break it down into cases.
Case 1: When the four people sit together in a block.
In this case, the 4 people can form a block and sit together in any of the 5 available spaces. Once the block is formed, the 4 people can arrange themselves within the block in 4! = 24 ways.
Total favorable arrangements in this case = 5 * 24 = 120
Case 2: When the four people sit with one empty seat between each person.
In this case, we can have 4 possible configurations where one person sits in the odd-numbered seats and the other person sits in the even-numbered seats. Each configuration can be arranged in 4! = 24 ways.
Total favorable arrangements in this case = 4 * 24 = 96
Step 3: Calculate the probability:
The probability of the favorable arrangements is given by the ratio of the number of favorable arrangements to the total number of arrangements.
Probability = (Total favorable arrangements) / (Total arrangements)
Probability = (120 + 96) / 1680
Probability = 216 / 1680
Probability = 1 / 7
Therefore, the correct answer is option C) 1/7.
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There are 8seats in row, where 4persons are to be seated. Then the probability of making them sit so that each person has exactly one neighbour, isa)1/14b)3/14c)1/7d)3/7Correct answer is option 'C'. Can you explain this answer?
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