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1.	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?
Answer» 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 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

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?

Answer»

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

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