Question

How many different words can be formed using all the letters of the word ${}^{\prime }COMBINATIO{N}^{\prime }$ such that the vowels as well as consonants appear in alphabetical order?

• A. $426$
• B. $462$
• C. $624$
• D. None of the above

Answer 1

The correct option is B $462$

Given letters $C,O,O,M,B,I,I,N,N,A,T$

Consonants $-C,M,B,N,N,T=6$ letters

Vowels $-O,O,I,I,A=5$ letters

Total permutations of the given word is equal to $\frac{11!}{2!2!2!}$

Total no. of arrangements of consonants $=\frac{6!}{2!}=360$

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 $=\frac{5!}{2!2!}=30$

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 $\frac{1}{360}\times \frac{1}{30}\times \frac{11!}{2!2!2!}=462$

Hence, the answer is $462.$