The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the…

The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is
  1. $36000$
  2. $37000$
  3. $34000$
  4. $35000$

Solution

DAUGHTER
Total words $=8$ !
Total words in which vowels are together $=6!\times 3!$ words in which all vowels are not together
$\begin{aligned}
& =8!-6!\times 3! \\ & =6![56-6] \\ & =720 \times 50 \\ & =36000
\end{aligned}$

Asked in: JEE Main 2025 (23 Jan Shift 1)

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