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Permutations and Combinations question

2004 · Shift 0 · Q103
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Permutations and Combinations question

2004 · Shift 0 · Q103

JEE MainMathematicsPermutations and CombinationsMCQ+4 / −1
How many ways are there to arrange the letters in the word GARDEN with vowels in alphabetical order
  1. A
    480
  2. B
    240
  3. C
    360
  4. D
    120
View written solutionFree

Correct answer: C

  1. The word GARDEN has 6 distinct letters: G,A,R,D,E,NG, A, R, D, E, NG,A,R,D,E,N

  2. The vowels in the word are: A,EA, EA,E

  3. The condition says the vowels must be in alphabetical order. Since alphabetical order of the vowels is: A before EA \text{ before } EA before E

  4. First, find the total number of arrangements of 6 distinct letters: 6!=7206! = 7206!=720

  5. In exactly half of these arrangements, AAA will appear before EEE, and in the other half, EEE will appear before AAA. Therefore, the number of arrangements with vowels in alphabetical order is: 6!2=7202=360\frac{6!}{2} = \frac{720}{2} = 36026!​=2720​=360

  6. Hence, the required number of arrangements is: 360360360

  7. Checking options:

    • A: 480480480
    • B: 240240240
    • C: 360360360 ✅
    • D: 120120120

Therefore, the correct option is C.

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