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In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?
- a)610
- b)720
- c)825
- d)920
Correct answer is option 'B'. Can you explain this answer?
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In how many different ways can the letters of the word 'OPTICAL' be ar...
The word 'OPTICAL' contains 7 different letters. When the vowels OIA are always together, they can be supposed to form one letter. Then, we have to arrange the letters PTCL (OIA). Now, 5 letters can be arranged in 5!=120 ways. The vowels (OIA) can be arranged among themselves in 3!=6 ways. Required number of ways =(120∗6)=720.
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In how many different ways can the letters of the word 'OPTICAL' be ar...
OPTICAL in this word we have 3 vowel. so write in rough what are those (o,i,a) Now how to solve , _p_t_c_l_ all the consonants we write and give the gap also then , how much gap in there?? 5 gap is here so we write it down, 5 factorial * how much vowel there (3 factorial) you know the factorial how to solve. 5 factorial * 3 factorial= (5*4*3*2*1) * (3*2*1) = 120 * 6 =720 (ans)
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In how many different ways can the letters of the word 'OPTICAL' be ar...
Solution:
To solve this problem, we need to first consider the vowels as a single unit. Therefore, we have:
- Vowel unit: OIA (3 letters)
- Consonant unit: PTL (3 letters)
Now, we need to arrange these two units in such a way that the vowels always come together. We can do this by treating the vowel unit as a single entity and arranging the 4-letter entity (including the vowel unit and one of the consonant letters) in different ways.
Step 1: Consider the 4-letter entity as a single unit
We have 4 letters in this entity:
- Vowel unit (OIA)
- One of the consonants (P or T or L)
Therefore, we can arrange these 4 letters in 4! = 24 ways.
Step 2: Arrange the vowel unit within the 4-letter entity
The vowel unit (OIA) can be arranged in 3! = 6 ways.
Step 3: Arrange the remaining consonants
After arranging the vowel unit within the 4-letter entity, we are left with two consonants (PTL) that need to be arranged. We can arrange these two consonants in 2! = 2 ways.
Step 4: Final arrangement
Multiplying the number of arrangements in each step, we get the total number of arrangements:
- 24 x 6 x 2 = 288
However, we need to remember that the vowel unit (OIA) can also be arranged in reverse order (AIO). Therefore, the total number of arrangements is:
- 288 x 2 = 576
Therefore, the correct answer is option B (720).
To solve this problem, we need to first consider the vowels as a single unit. Therefore, we have:
- Vowel unit: OIA (3 letters)
- Consonant unit: PTL (3 letters)
Now, we need to arrange these two units in such a way that the vowels always come together. We can do this by treating the vowel unit as a single entity and arranging the 4-letter entity (including the vowel unit and one of the consonant letters) in different ways.
Step 1: Consider the 4-letter entity as a single unit
We have 4 letters in this entity:
- Vowel unit (OIA)
- One of the consonants (P or T or L)
Therefore, we can arrange these 4 letters in 4! = 24 ways.
Step 2: Arrange the vowel unit within the 4-letter entity
The vowel unit (OIA) can be arranged in 3! = 6 ways.
Step 3: Arrange the remaining consonants
After arranging the vowel unit within the 4-letter entity, we are left with two consonants (PTL) that need to be arranged. We can arrange these two consonants in 2! = 2 ways.
Step 4: Final arrangement
Multiplying the number of arrangements in each step, we get the total number of arrangements:
- 24 x 6 x 2 = 288
However, we need to remember that the vowel unit (OIA) can also be arranged in reverse order (AIO). Therefore, the total number of arrangements is:
- 288 x 2 = 576
Therefore, the correct answer is option B (720).
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In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?a)610b)720c)825d)920Correct answer is option 'B'. Can you explain this answer?
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