The total number of ways the word 'DAUGHTER' can be arranged so that all vowels don't occur together:

Question

Let \(y=y(x)\) be the solution of the differential equation \[ x\frac{dy}{dx}=y-x^2\cot x,\quad x\in(0,\pi) \] If \(y\!\left(\frac{\pi}{2}\right)=\frac{\pi^2}{2}\), then \[ 6y\!\left(\frac{\pi}{6}\right)-8y\!\left(\frac{\pi}{4}\right) \] is equal to:

Answer 1

To arrange the letters of "DAUGHTER" such that vowels are not adjacent:

Arrange the 5 consonants (D, G, H, T, R) in \( 5! = 120 \) ways.
Create 6 possible positions for the 3 vowels (A, U, E) around the consonants. Choose 3 of these positions in \( \binom{6}{3} = 20 \) ways.
Arrange the 3 vowels within the chosen positions in \( 3! = 6 \) ways.

The total number of arrangements is \( 5! \times \binom{6}{3} \times 3! = 120 \times 20 \times 6 = 36000 \). The correct answer is option (1).

The total number of ways the word 'DAUGHTER' can be arranged so that all vowels don't occur together:

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The Correct Option is C

Solution and Explanation

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