In how many different ways can the letters of | vedclass

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Question

(D) The word '$THERAPY$' consists of $7$ distinct letters: $T, H, E, R, A, P, Y$.There are $2$ vowels $(E, A)$ and $5$ consonants $(T, H, R, P, Y)$.Fi

Vedclass · Accepted Answer

$3600$

Vedclass · Answer

(D) The word '$THERAPY$' consists of $7$ distinct letters: $T, H, E, R, A, P, Y$.There are $2$ vowels $(E, A)$ and $5$ consonants $(T, H, R, P, Y)$.First,calculate the total number of arrangements of all $7$ letters,which is $7! = 5040$.Next,calculate the number of arrangements where the $2$ vowels come together. Treat the $2$ vowels as a single unit. Now we have $5$ consonants + $1$ unit = $6$ entities,which can be arranged in $6!$ ways.The $2$ vowels within the unit can be arranged in $2!$ ways.So,the number of ways the vowels come together is $6! 	imes 2! = 720 	imes 2 = 1440$.The number of ways the vowels never come together is the total arrangements minus the arrangements where they are together:$5040 - 1440 = 3600$.

In how many different ways can the letters of the word '$THERAPY$' be arranged so that the vowels never come together?

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