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There are 7 keys in a key ring. If two more keys are to be added in the ring at random, what is the probability that two keys will be adjacent?
- a)1/8
- b)1/6
- c)1/5
- d)1/4
- e)1/3
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
There are 7 keys in a key ring. If two more keys are to be added in th...
Understanding the Problem
We have a key ring with 7 keys, and we want to add 2 more keys at random. We need to find the probability that these two new keys will be adjacent.
Possible Arrangements
- The total number of keys after adding the 2 new keys will be 9.
- When arranging these keys in a circle, the number of distinct arrangements is given by (n-1)!, where n is the total number of keys.
- Therefore, the number of arrangements for 9 keys is 8!.
Adjacent Keys Calculation
- To count the arrangements where the two new keys are adjacent, we can treat the two new keys as a single unit or block.
- This creates a new scenario with 8 units (the block of 2 keys + 7 original keys).
- The number of distinct arrangements for these 8 units in a circle is 7!.
Probability Calculation
- The probability that the two new keys are adjacent is the ratio of the number of arrangements with adjacent keys to the total arrangements.
- Probability = (Number of adjacent arrangements) / (Total arrangements) = 7! / 8! = 7 / 8.
Final Step
- However, since there are 2 ways to arrange the 2 new keys within their block (Key A next to Key B or Key B next to Key A), we multiply the adjacent arrangements by 2.
- Therefore, the effective number of adjacent arrangements becomes 2 * 7!.
- Now, the probability becomes (2 * 7!) / 8! = 2 / 8 = 1 / 4.
Thus, the correct probability that the two keys will be adjacent is 1/4 (option D).
We have a key ring with 7 keys, and we want to add 2 more keys at random. We need to find the probability that these two new keys will be adjacent.
Possible Arrangements
- The total number of keys after adding the 2 new keys will be 9.
- When arranging these keys in a circle, the number of distinct arrangements is given by (n-1)!, where n is the total number of keys.
- Therefore, the number of arrangements for 9 keys is 8!.
Adjacent Keys Calculation
- To count the arrangements where the two new keys are adjacent, we can treat the two new keys as a single unit or block.
- This creates a new scenario with 8 units (the block of 2 keys + 7 original keys).
- The number of distinct arrangements for these 8 units in a circle is 7!.
Probability Calculation
- The probability that the two new keys are adjacent is the ratio of the number of arrangements with adjacent keys to the total arrangements.
- Probability = (Number of adjacent arrangements) / (Total arrangements) = 7! / 8! = 7 / 8.
Final Step
- However, since there are 2 ways to arrange the 2 new keys within their block (Key A next to Key B or Key B next to Key A), we multiply the adjacent arrangements by 2.
- Therefore, the effective number of adjacent arrangements becomes 2 * 7!.
- Now, the probability becomes (2 * 7!) / 8! = 2 / 8 = 1 / 4.
Thus, the correct probability that the two keys will be adjacent is 1/4 (option D).
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Community Answer
There are 7 keys in a key ring. If two more keys are to be added in th...
Let the keys be name as A B C D E F G H I [H and I being the two new keys]
Probability of H and I being together = 
Total number of arrangements possible in a circle with the 9 keys = (n-1)! = 8!
Total number of arrangements where H and I are together = 7! x 2 [Consider H and I as one unit. In addition, there are 7 other units i.e. A B C D E F G. Total number of ways to arrange the 8 units in a circle is (n-1)! = 7!. Further, H and I can be arranged in 2 ways i.e. H I or I H.]
Probability = (7! x 2)/8! = 1/4.
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There are 7 keys in a key ring. If two more keys are to be added in the ring at random, what is the probability that two keys will be adjacent?a)1/8b)1/6c)1/5d)1/4e)1/3Correct answer is option 'D'. Can you explain this answer?
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