The number of six-letter words (with or witho | vedclass

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Question

(A) The word $VOWELS$ contains $6$ distinct letters: $V, O, W, E, L, S$.There are $2$ vowels $(O, E)$ and $4$ consonants $(V, W, L, S)$.Total number o

Vedclass · Accepted Answer

$576$

Vedclass · Answer

(A) The word $VOWELS$ contains $6$ distinct letters: $V, O, W, E, L, S$.There are $2$ vowels $(O, E)$ and $4$ consonants $(V, W, L, S)$.Total number of arrangements of all $6$ letters is $6! = 720$.To find the number of arrangements where all consonants come together,we treat the $4$ consonants as a single unit. This unit along with the $2$ vowels gives $1 + 2 = 3$ items to arrange,which can be done in $3!$ ways.The $4$ consonants within the unit can be arranged among themselves in $4!$ ways.So,the number of arrangements where all consonants are together is $3! 	imes 4! = 6 	imes 24 = 144$.The number of arrangements where all consonants never come together is (Total arrangements) - (Arrangements where all consonants are together).$= 720 - 144 = 576$.

The number of six-letter words (with or without meaning) formed using all the letters of the word $VOWELS$ such that all the consonants never come together is ... .

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