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 =

Let C and T be denoted by a single letter Z

New word is APAINZ

This can be permuted in = 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