How many combination of letters can be formed using all the letters of
the word 'COALSHED' so that the vowels are always together?
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Explanation:
Given letters C,O,O,M,B,I,I,N,N,A,T
Consonants −C,M,B,N,N,T=6 letters
Vowels −O,O,I,I,A=5 letters
Total permutations of the given word is equal to
2!2!2!
11!
Total no. of arrangements of consonants =
2!
6!
=360
Out of these 360 ways , only one way has the alphabets in the order B,C,M,N,N,T ( alphabetical )
Similarily for vowels total =
2!2!
5!
=30
Only one of 30 has the alphabets in the order A,I,I,O,O
∴ By symmetry the arrangements of given word with consonants and vowels in alphabetical order is
360
1
×
30
1
×
2!2!2!
11!
=462
Hence, the answer is 462.
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