$TESTBOOK$
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
$\qquad = 5\times 4\times 3 \times \dfrac{6!}{3!\times 3!} \times \dfrac{3!}{2}$
$\qquad =\dfrac{5\times 4\times 3 \times 6\times 5\times 4}{2}$
$\qquad =\dfrac{7200}{2}$
$\qquad=\text{3600 ways}$