  Question

How many different words can be formed using all the letters of the word $$'COMBINATION'$$ such that the vowels as well as consonants appear in alphabetical order?

A
426  B
462  C
624  D
None of the above  Solution

The correct option is B $$462$$Given letters $$C,O,O,M,B,I,I,N,N,A,T$$Consonants $$- \;C,M,B,N,N,T=6$$ lettersVowels $$-\;O,O,I,I,A=5$$ lettersTotal permutations of the given word is equal to $$\dfrac{11!}{2!2!2!}$$Total no. of arrangements of consonants $$=\dfrac{6!}{2!}=360$$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 $$=\dfrac{5!}{2!2!}=30$$Only one of $$30$$ has the alphabets in the order $$A,I,I,O,O$$$$\therefore$$ By symmetry the arrangements of given word with consonants and vowels in alphabetical order is $$\dfrac{1}{360}\times \dfrac{1}{30} \times \dfrac{11!}{2!2!2!}=462$$Hence, the answer is $$462.$$Maths

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