Question:
How many different words can be formed with the letters of the word ‘CAPTAIN’? In how many of these C and T are never together?
Solution:
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