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In how many different ways can the letters of the word 'LEADING' be arranged such that the vowels should always come together?
- a)122
- b)720
- c)420
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
In how many different ways can the letters of the word 'LEADING' be ar...
The word 'LEADING' has 7 different letters.
When the vowels EAI are always together, they can be supposed to form one letter.
Then, we have to arrange the letters LNDG (EAI).
Now, 5 (4 + 1) letters can be arranged in 5! = 120 ways.
The vowels (EAI) can be arranged among themselves in 3! = 6 ways.
Required number of ways = (120 * 6) = 720.
Most Upvoted Answer
In how many different ways can the letters of the word 'LEADING' be ar...
LEADING has 3vowels and 4 consonants. Keeping all vowels together LDNG(EAI) Hence vowels always coming together will be: 5¡ *3¡ It will be 720.
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In how many different ways can the letters of the word 'LEADING' be ar...
To solve this problem, we need to consider the vowels (EAI) as one group and treat it as a single letter. The word 'LEADING' has 7 letters, including 3 vowels (EAI) and 4 consonants (LNDG).
To find the number of arrangements where the vowels come together, we can consider the group of vowels (EAI) as a single letter. This reduces the problem to arranging 5 letters (the vowel group and the consonants) instead of 7.
To find the number of arrangements, we can use the formula for arranging n objects with k indistinguishable objects, which is n! / (k! * (n-k)!).
Let's calculate the number of arrangements step by step:
Step 1: Consider the group of vowels (EAI) as a single letter.
- We now have 5 letters to arrange: LNDG and the vowel group (EAI).
Step 2: Calculate the number of arrangements of the 5 letters.
- The number of arrangements is 5! / (2! * 3!) = (5 * 4 * 3!) / (2 * 3!) = 5 * 4 = 20.
Step 3: Consider the arrangements within the vowel group.
- Since the vowels (EAI) are considered as a single letter, the vowel group can be arranged among themselves in 3! = 6 ways.
Step 4: Multiply the number of arrangements from step 2 and step 3.
- The total number of arrangements where the vowels come together is 20 * 6 = 120.
Therefore, there are 120 different ways to arrange the letters of the word 'LEADING' such that the vowels come together.
Hence, the correct answer is option B) 720.
To find the number of arrangements where the vowels come together, we can consider the group of vowels (EAI) as a single letter. This reduces the problem to arranging 5 letters (the vowel group and the consonants) instead of 7.
To find the number of arrangements, we can use the formula for arranging n objects with k indistinguishable objects, which is n! / (k! * (n-k)!).
Let's calculate the number of arrangements step by step:
Step 1: Consider the group of vowels (EAI) as a single letter.
- We now have 5 letters to arrange: LNDG and the vowel group (EAI).
Step 2: Calculate the number of arrangements of the 5 letters.
- The number of arrangements is 5! / (2! * 3!) = (5 * 4 * 3!) / (2 * 3!) = 5 * 4 = 20.
Step 3: Consider the arrangements within the vowel group.
- Since the vowels (EAI) are considered as a single letter, the vowel group can be arranged among themselves in 3! = 6 ways.
Step 4: Multiply the number of arrangements from step 2 and step 3.
- The total number of arrangements where the vowels come together is 20 * 6 = 120.
Therefore, there are 120 different ways to arrange the letters of the word 'LEADING' such that the vowels come together.
Hence, the correct answer is option B) 720.
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