The total number of words that can be formed | vedclass

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Question

(C) The word $ALLEN$ contains $5$ letters: $A, L, L, E, N$. The vowels are $A$ and $E$. The total number of arrangements of the letters $A, L, L, E, N

Vedclass · Accepted Answer

$30$

Vedclass · Answer

(C) The word $ALLEN$ contains $5$ letters: $A, L, L, E, N$. The vowels are $A$ and $E$. The total number of arrangements of the letters $A, L, L, E, N$ without any restriction is $\frac{5!}{2!} = \frac{120}{2} = 60$. In these $60$ arrangements,the vowels $A$ and $E$ can appear in $2! = 2$ relative orders ($AE$ or $EA$). Since we require the vowels to appear in alphabetical order ($A$ before $E$),we only consider the cases where they appear as $AE$. Therefore,the required number of words is $\frac{60}{2!} = 30$.

The total number of words that can be formed by using all the letters of the word $ALLEN$ such that all vowels occur in alphabetical order is:

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