JEE Main 2025 Jan 23 Shift 1, Mathematics Q12: Permutation

JEE Main2025Jan 23, Shift 1Mathematics

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

Solution

DAUGHTER has 8 distinct letters with 3 vowels (A, U, E) and 5 consonants.

Total arrangements: $8! = 40320$ .

Arrangements with all three vowels together (treat them as a single block): $6! \times 3! = 720 \times 6 = 4320$ .

Arrangements where vowels are NOT all together: $40320 - 4320 = 36000$ .

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