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Find the number of permutations of the letters of the word UNIQUE 

(a) How many of them end with QUE. 

(b) How many begin with U and end with E. 

(c) In how many vowels are in even places. 

(d) In how many all vowels are together.

1 Answer

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There are six letters in which repeated twice can be permutted in \(\frac{6!}{2!}\) ways \(\frac{720}{2}\) = 360 ways 

(a) The words end with QUE: 

The remaining 3 letters can be arranged in 3! Ways = 6 ways. 

(b) The words begin with U and end with E: 

The remaining 4 letters can be permuted in 4! Ways = 24 ways. 

(c) The four vowels can be placed in 3 even places in 4P3, ways and the remaining 3 can be permutted in 3! ways i.e., \(\frac{4 \times 3\times 2}{2}\)= 12 ways. 

(d) All 4 vowels can be taken as 1 unit 

∴ the total 1 + 2 = 3 can be done in 3! Ways and 4 vowels can be permuted in \(\frac{4!}{2!}\)= 12 ways. 

∴ The number of ways = 12 × 6 = 72 ways.

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