Find the number of different 8-letter arrange | vedclass

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Question

(A) The word $DAUGHTER$ contains $8$ distinct letters: $D, A, U, G, H, T, E, R$. The vowels are $A, U, E$ ($3$ vowels) and the consonants are $D, G, H

Vedclass · Accepted Answer

$36000$

Vedclass · Answer

(A) The word $DAUGHTER$ contains $8$ distinct letters: $D, A, U, G, H, T, E, R$. The vowels are $A, U, E$ ($3$ vowels) and the consonants are $D, G, H, T, R$ ($5$ consonants).Total arrangements of $8$ letters $= 8! = 40320$.To find the arrangements where all vowels are together,treat $(A, U, E)$ as a single unit. Now we have $5$ consonants $+ 1$ unit $= 6$ items,which can be arranged in $6!$ ways. The $3$ vowels within the unit can be arranged in $3!$ ways.Number of arrangements where vowels are together $= 6! 	imes 3! = 720 	imes 6 = 4320$.Number of arrangements where vowels do not occur together $=$ Total arrangements $-$ Arrangements where vowels are together.$= 40320 - 4320 = 36000$.

Find the number of different $8$-letter arrangements that can be made from the letters of the word $DAUGHTER$ such that all vowels do not occur together.

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