In how many ways can the letters of the word | vedclass

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Question

(C) The word $GARDEN$ has $6$ distinct letters: $G, A, R, D, E, N$.The vowels in the word are $A$ and $E$.Total number of arrangements of $6$ letters

Vedclass · Accepted Answer

$360$

Vedclass · Answer

(C) The word $GARDEN$ has $6$ distinct letters: $G, A, R, D, E, N$.The vowels in the word are $A$ and $E$.Total number of arrangements of $6$ letters is $6! = 720$.In any arrangement,the vowels $A$ and $E$ can appear in $2! = 2$ ways: $(A, E)$ or $(E, A)$.Since we require the vowels to appear in alphabetical order,we only consider the case where $A$ comes before $E$.By symmetry,exactly half of the total arrangements will have $A$ before $E$,and the other half will have $E$ before $A$.Therefore,the number of arrangements where $A$ appears before $E$ is $\frac{6!}{2!} = \frac{720}{2} = 360$.

In how many ways can the letters of the word $GARDEN$ be arranged such that the vowels appear in alphabetical order?

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