How many words can be acquired by the word universal, Computer Engineering

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How many words can be acquired by arranging the letters of the word 'UNIVERSAL' in different way? In how many of them 

(i)  E, R, S takes place together

(ii)  No two of the letters E, R, S take place together.   

Ans: There are nine letters in the word 'UNIVERSAL' and none is repeated. So they are permuted in 9! ways.    

(i)  While E, R, S occur together, there will be 6 other letters and one (ERS). They are permuted in 7! Ways and so there will be 7! * 3! Words comprising E,R,S together.

(ii) Any two letters of E, R, S can come together in 3P2 = 6 ways. While two letters (out of 9) come together after that 7 +1 = 8 letters can be arranged in 8! ways. So there will be 8!* 6 words in which any two of E, R, S come together.

 ∴Number of words in which no two of E,R,S come together = 9! - 8!*6


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