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The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is ________.
Correct answer is '12'. Can you explain this answer?
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The number of permutations of the characters in LILAC so that no chara...
Since both L’s are indistinguishable.
First L’s can be arranged in 3 positions 2, 3, or 5 in 3C2 = 3 ways as follows:
Now the letters I, A, C can be deranged in 2 × 2! ways. Example in
C cannot occupy 5th position, so only 2 ways.
Remaining I and A can be arranged in remaining 2 position in 2! ways = 2 ways.
So answer is 3 × 2 × 2! = 12.
First L’s can be arranged in 3 positions 2, 3, or 5 in 3C2 = 3 ways as follows:
Now the letters I, A, C can be deranged in 2 × 2! ways. Example in
C cannot occupy 5th position, so only 2 ways.Remaining I and A can be arranged in remaining 2 position in 2! ways = 2 ways.
So answer is 3 × 2 × 2! = 12.
Most Upvoted Answer
The number of permutations of the characters in LILAC so that no chara...
To solve this problem, we can use the principle of inclusion-exclusion.
First, let's consider the case where we don't have any restrictions. The number of permutations of the characters in LILAC is 5!, which is equal to 120.
Now, let's consider the case where at least one of the L's appears in its original position. There are 2 L's in LILAC, so we have two possibilities:
1. The first L is in its original position and the second L is not.
2. The second L is in its original position and the first L is not.
For the first case, we can fix the first L in its original position and permute the remaining 4 characters (I, L, A, C) in 4! ways. For the second case, we can fix the second L in its original position and permute the remaining 4 characters in 4! ways as well.
However, we have counted the case where both L's are in their original positions twice. So, we need to subtract this case from the total.
Therefore, the number of permutations where at least one L appears in its original position is 2 * 4! - 3! = 48 - 6 = 42.
Finally, the number of permutations where no character appears in its original position is the total number of permutations minus the number of permutations where at least one L appears in its original position:
120 - 42 = 78.
So, there are 78 permutations of the characters in LILAC where no character appears in its original position.
First, let's consider the case where we don't have any restrictions. The number of permutations of the characters in LILAC is 5!, which is equal to 120.
Now, let's consider the case where at least one of the L's appears in its original position. There are 2 L's in LILAC, so we have two possibilities:
1. The first L is in its original position and the second L is not.
2. The second L is in its original position and the first L is not.
For the first case, we can fix the first L in its original position and permute the remaining 4 characters (I, L, A, C) in 4! ways. For the second case, we can fix the second L in its original position and permute the remaining 4 characters in 4! ways as well.
However, we have counted the case where both L's are in their original positions twice. So, we need to subtract this case from the total.
Therefore, the number of permutations where at least one L appears in its original position is 2 * 4! - 3! = 48 - 6 = 42.
Finally, the number of permutations where no character appears in its original position is the total number of permutations minus the number of permutations where at least one L appears in its original position:
120 - 42 = 78.
So, there are 78 permutations of the characters in LILAC where no character appears in its original position.
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The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is ________.Correct answer is '12'. Can you explain this answer?
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The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is ________.Correct answer is '12'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is ________.Correct answer is '12'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is ________.Correct answer is '12'. Can you explain this answer?.
The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is ________.Correct answer is '12'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is ________.Correct answer is '12'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is ________.Correct answer is '12'. Can you explain this answer?.
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