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Question

In how many ways can the letters of the word 'STRANGE' be arranged so that 
(i) the vowels come together? 
(ii) the vowels never come together? and 
(iii) the vowels occupy only the odd places?


Solution

There are 7 letters in the word. STRANGE, including 2 vowels (A, E) and 5 consonants (S, T, R, N, G). 
(i) Considering 2 vowels as one letter, we have 6 letters which can be arranged in 6P6 = 6! ways A, E can be put together in 2! ways. 
Hence, required number of words 
= 6!×2!
= 6×5×4×3×2×1×2
= 720×2
= 1440.
(ii) The total number of words formed by.: using all the letters of the words 'STRANGE'is 7P7=7!
= 7×6×5×4×3×2×1
= 5040.
So, the total number of words in which vowels are never together 
= Total number of words - number of words in which vowels are always together
= 5040-1440 
=3600
(iii) There are 7 letters in the word 'STRANGE', out of these letters 'A' and `E' are the vowels. 
There are 4 odd places in the word 'STRANGE'. The two vowels can be arrangd in 4P2 ways. 
The remaining 5 consonants can be arranged among themselves in 5P5 ways. 
The total number of arrangements. 
= 4P2×5P5
= 4!2!×5!
= 1440

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