How many different words can be formed with the letters of the word

To find: number of words such that C and T are never together

Number of words where C and T are never the together = Total numbers of words -

Number of words where C and T are together

Total number of words $=\frac{7 !}{2 !}=2520$

Let C and T be denoted by a single letter Z

$\Rightarrow$ New word is APAINZ

This can be permuted in $\frac{6 !}{2 !}=360$ ways

Z can be permuted among itself in 2 ways

⇒ Number of words where C and T are together = 360 × 2 = 720

⇒ Number of words where C and T are never together = 2520 - 720 = 1800

There are 1800 words where C and T are never together