In how many different ways can the letters of the word "DETERRANT" be arranged so that the repeated letters do not come together?

Solution:
Total number of Letters = 9.
Number of E's = 2
Number of T's = 2
Number of R's = 2
Thus, Total Number of arrangements = $\frac{9!}{2! \times 2! \times 2!} = 45360$
Now, Total Number of Arrangements if repeated terms come together = $4! \times \frac{6!}{2! \times 2! \times 2!} = 2160$
Thus, Total Arrangements so that the repeated letters do not come together = Total Arrangements - Total Number of Arrangements if repeated terms come together
$= 45360 - 2160$
$= 43200$