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The letters of the word 'BEAUTY' are arranged such that the following conditions are met. How many arrangements are possible?
Condition 1: The vowels should come in the same order as in the original word.
Condition 2: No 2 consonants should come together.
Condition 1: The vowels should come in the same order as in the original word.
Condition 2: No 2 consonants should come together.
- a)4
- b)4.3!
- c)4.(3!)(3!)
- d)3!
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
The letters of the word BEAUTY are arranged such that the following co...
The word "BEAUTY" has 6 letters, out of which 3 are vowels (E, A, U) and 3 are consonants (B, T, Y). We need to arrange the letters in such a way that the vowels come in the same order as in the original word and no two consonants are together.
To solve this problem, we can break it down into two steps:
Step 1: Arrange the vowels in the same order as in the original word.
Step 2: Arrange the consonants in such a way that no two consonants are together.
Step 1: Arranging the vowels
Since the vowels (E, A, U) need to be arranged in the same order, we can consider them as a single entity. So, we have 1 entity (EAU) and 3 consonants (B, T, Y) to arrange.
Step 2: Arranging the consonants
To ensure that no two consonants are together, we can use the concept of permutations with restrictions. We can place the 3 consonants in the 4 available gaps between the vowels.
_ E _ A _ U _
Since there are 4 gaps, we can place the first consonant (B) in any of the 4 gaps. After placing the first consonant, we will have 3 gaps left.
_ E B A _ U _
Now, we can place the second consonant (T) in any of the 3 remaining gaps.
_ E B A T U _
Finally, we can place the third consonant (Y) in any of the 2 remaining gaps.
_ E B A T U Y
Thus, we have arranged the consonants in such a way that no two consonants are together. Now, we need to calculate the number of ways to arrange the vowels and the consonants separately.
For the vowels (EAU), there is only 1 way to arrange them since they need to be in the same order.
For the consonants (BTY), there are 3! = 3 * 2 * 1 = 6 ways to arrange them.
Therefore, the total number of arrangements satisfying the given conditions is 1 * 6 = 6.
Hence, the correct answer is option B) 6.
To solve this problem, we can break it down into two steps:
Step 1: Arrange the vowels in the same order as in the original word.
Step 2: Arrange the consonants in such a way that no two consonants are together.
Step 1: Arranging the vowels
Since the vowels (E, A, U) need to be arranged in the same order, we can consider them as a single entity. So, we have 1 entity (EAU) and 3 consonants (B, T, Y) to arrange.
Step 2: Arranging the consonants
To ensure that no two consonants are together, we can use the concept of permutations with restrictions. We can place the 3 consonants in the 4 available gaps between the vowels.
_ E _ A _ U _
Since there are 4 gaps, we can place the first consonant (B) in any of the 4 gaps. After placing the first consonant, we will have 3 gaps left.
_ E B A _ U _
Now, we can place the second consonant (T) in any of the 3 remaining gaps.
_ E B A T U _
Finally, we can place the third consonant (Y) in any of the 2 remaining gaps.
_ E B A T U Y
Thus, we have arranged the consonants in such a way that no two consonants are together. Now, we need to calculate the number of ways to arrange the vowels and the consonants separately.
For the vowels (EAU), there is only 1 way to arrange them since they need to be in the same order.
For the consonants (BTY), there are 3! = 3 * 2 * 1 = 6 ways to arrange them.
Therefore, the total number of arrangements satisfying the given conditions is 1 * 6 = 6.
Hence, the correct answer is option B) 6.
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Community Answer
The letters of the word BEAUTY are arranged such that the following co...
First considering Condition 2 that no 2 consonants should come together, we arrange the vowels in the following way:
_ V _ V _ V _
Hence the consonants can be placed in any 3 of the 4 gaps, to ensure they do not come together.
Condition 1 says that the vowels are in the same order as in the original word, that is, EAU, so we can put them in the arrangement in only 1 way, that is,
_ E _ A _ U _
Now the 4 remaining places can be filled by 3 consonants in 4C3=4 ways.
Now, since the consonants can rearrange among themselves in all possible ways, number of rearrangements = 3!.
Hence, total number of arrangements possible = 4.3!
_ V _ V _ V _
Hence the consonants can be placed in any 3 of the 4 gaps, to ensure they do not come together.
Condition 1 says that the vowels are in the same order as in the original word, that is, EAU, so we can put them in the arrangement in only 1 way, that is,
_ E _ A _ U _
Now the 4 remaining places can be filled by 3 consonants in 4C3=4 ways.
Now, since the consonants can rearrange among themselves in all possible ways, number of rearrangements = 3!.
Hence, total number of arrangements possible = 4.3!
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The letters of the word BEAUTY are arranged such that the following conditions are met. How many arrangements are possible?Condition 1: The vowels should come in the same order as in the original word.Condition 2: No 2 consonants should come together.a)4b)4.3!c)4.(3!)(3!)d)3!Correct answer is option 'B'. Can you explain this answer?
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