Question:
In how many different ways can the letter of the word 'EMBROIDERY' be arranged so that all vowels are always positioned together?
In how many different ways can the letter of the word 'EMBROIDERY' be arranged so that all vowels are always positioned together?
Updated On: Sep 16, 2026
- 1,20,960
- 8,640
- 4,320
- 30,240
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The Correct Option is D
Solution and Explanation
Concept:
Step 1: List the letters of 'EMBROIDERY' and separate vowels from consonants
Letters: E, M, B, R, O, I, D, E, R, Y (10 letters total)
Vowels: E, O, I, E → 4 vowels (E repeats twice)
Consonants: M, B, R, D, R, Y → 6 consonants (R repeats twice)
Step 2: Treat all 4 vowels as a single block and arrange the units
Units to arrange = 6 consonants + 1 vowel-block = 7 units
Since R repeats twice among the consonants, arrangements of these 7 units = 7!/2! = 5040/2 = 2520
Step 3: Arrange the vowels within their block
Vowels are E, E, I, O (E repeats twice)
Arrangements within block = 4!/2! = 24/2 = 12
Step 4: Multiply the two results
Total arrangements = 2520 x 12 = 30,240
Final Answer: 30,240
- When certain letters must always stay together, tie them into a single block and arrange this block along with the remaining letters.
- Multiply by the number of ways to arrange the letters within that block.
- Divide by the factorial of the count of each repeated letter to avoid overcounting identical arrangements.
Step 1: List the letters of 'EMBROIDERY' and separate vowels from consonants
Letters: E, M, B, R, O, I, D, E, R, Y (10 letters total)
Vowels: E, O, I, E → 4 vowels (E repeats twice)
Consonants: M, B, R, D, R, Y → 6 consonants (R repeats twice)
Step 2: Treat all 4 vowels as a single block and arrange the units
Units to arrange = 6 consonants + 1 vowel-block = 7 units
Since R repeats twice among the consonants, arrangements of these 7 units = 7!/2! = 5040/2 = 2520
Step 3: Arrange the vowels within their block
Vowels are E, E, I, O (E repeats twice)
Arrangements within block = 4!/2! = 24/2 = 12
Step 4: Multiply the two results
Total arrangements = 2520 x 12 = 30,240
Final Answer: 30,240
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