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NUMBER OF WAYS WE CAN ARRAYS LETTERS OF THE WORD "TESTBOOK" SO THAT NO TWO VOWELS ARE TOGETHER IS
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$TESTBOOK$

In this word $T$ appears $2$ time

$E$ appears $1$ time

$S$ appears $1$ time

$B$ appears $1$ time

$O$ appears $2$ time

$K$ appears $1$ time

∴ There are $6$ letter, where $2$ letters are vowel & $4$ are non-vowel

& it's a $8$ letter-word.

Without vowel we have $4$ letters in which $1$ letter appears $2$ time.

So, we have total $5$ places where we can place the non-vowel letter.

This non-vowel can itself arrange in $\dfrac{5!}{2}$ ways $\qquad \big[∵ \text{T appears 2 times}\big]$

Between, the non-vowels, there are $6$ places.

∴ We can place the vowels in these $6$ places & we have $3$ vowels.

∴ We can choose & arrange $3$ vowels into these $6$ places in $^6C_3 \times \dfrac{3!}{2}$ ways $^6C_3\rightarrow\text{choose 3 places from 6 places to place 3 vowels}\\\dfrac{3!}{2}\rightarrow \text{arrange 3 vowels & divide by 2 because O appears twice}$

∴ Total no. of ways to arrange the letter of the word "TESTBOOK" so that no two vowels are together= $\dfrac{5!}{2} \times$ $^6C_3 \times \dfrac{3!}{2} $ ways

$\qquad = 5\times 4\times 3 \times \dfrac{6!}{3!\times 3!} \times \dfrac{3!}{2}$

$\qquad =\dfrac{5\times 4\times 3 \times 6\times 5\times 4}{2}$

$\qquad =\dfrac{7200}{2}$

$\qquad=\text{3600 ways}$

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