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How many total words can be formed from the letters of the word INSURANCE in which vowels are always together?
- a)18270
- b)8640
- c)12780
- d)12790
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
How many total words can be formed from the letters of the word INSURA...
Solution:
Given, the word is INSURANCE.
We need to find the total number of words that can be formed from the given word, in which vowels are always together.
Let's first look at the total number of ways in which the letters of the word INSURANCE can be arranged.
Total number of letters in the given word = 9
Number of times each letter appears in the given word:
- I appears twice
- N, S, U, R, A, and C appear once each
Therefore, the total number of ways in which the letters of the word INSURANCE can be arranged is given by:
= 9! / (2! * 1! * 1! * 1! * 1! * 1! * 1!)= 362880 / 2= 181440
Now, let's consider the vowels (I, U, and E) as a single letter and arrange them first. This can be done in 3! ways.
Now we have 7 letters, including the "vowel letter," to be arranged. This can be done in 7! / (2! * 1! * 1! * 1! * 1! * 1! * 1!) ways.
Therefore, the total number of words that can be formed from the letters of the word INSURANCE in which vowels are always together is given by:
3! * 7! / (2! * 1! * 1! * 1! * 1! * 1! * 1!) = 6 * 5040 / 2 = 30240
Hence, the correct option is (B) 8640.
Given, the word is INSURANCE.
We need to find the total number of words that can be formed from the given word, in which vowels are always together.
Let's first look at the total number of ways in which the letters of the word INSURANCE can be arranged.
Total number of letters in the given word = 9
Number of times each letter appears in the given word:
- I appears twice
- N, S, U, R, A, and C appear once each
Therefore, the total number of ways in which the letters of the word INSURANCE can be arranged is given by:
= 9! / (2! * 1! * 1! * 1! * 1! * 1! * 1!)= 362880 / 2= 181440
Now, let's consider the vowels (I, U, and E) as a single letter and arrange them first. This can be done in 3! ways.
Now we have 7 letters, including the "vowel letter," to be arranged. This can be done in 7! / (2! * 1! * 1! * 1! * 1! * 1! * 1!) ways.
Therefore, the total number of words that can be formed from the letters of the word INSURANCE in which vowels are always together is given by:
3! * 7! / (2! * 1! * 1! * 1! * 1! * 1! * 1!) = 6 * 5040 / 2 = 30240
Hence, the correct option is (B) 8640.
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Community Answer
How many total words can be formed from the letters of the word INSURA...
The word Imsurance has 9 Words in which I U A and E are vowels and the other letters are consonants.
Now the vowels are to occur together always so consider all the vowels as 1 object and there are 5 consonants remaining
So we have 6 objects In total (IUAE) N S R C N
N is repeated twice so
The total no.of ways=n!/p!=6!/2!=360
Now the 4 vowels can be arranged in themselves in ⁴P4 ways = 4! ways=24
Total number of words that can be formed=
360×24=8640
Hope this helps!
Happy learning!
Now the vowels are to occur together always so consider all the vowels as 1 object and there are 5 consonants remaining
So we have 6 objects In total (IUAE) N S R C N
N is repeated twice so
The total no.of ways=n!/p!=6!/2!=360
Now the 4 vowels can be arranged in themselves in ⁴P4 ways = 4! ways=24
Total number of words that can be formed=
360×24=8640
Hope this helps!
Happy learning!
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How many total words can be formed from the letters of the word INSURANCE in which vowels are always together?a)18270b)8640c)12780d)12790Correct answer is option 'B'. Can you explain this answer?
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How many total words can be formed from the letters of the word INSURANCE in which vowels are always together?a)18270b)8640c)12780d)12790Correct answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about How many total words can be formed from the letters of the word INSURANCE in which vowels are always together?a)18270b)8640c)12780d)12790Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for How many total words can be formed from the letters of the word INSURANCE in which vowels are always together?a)18270b)8640c)12780d)12790Correct answer is option 'B'. Can you explain this answer?.
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