How many ways are there to arrange the letter | vedclass

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Question

(A) The word $EDUCATION$ has $9$ letters: $5$ vowels $(E, U, A, I, O)$ and $4$ consonants $(D, C, T, N)$.
First,arrange the $5$ vowels in the given o

Vedclass · Accepted Answer

$15$

Vedclass · Answer

(A) The word $EDUCATION$ has $9$ letters: $5$ vowels $(E, U, A, I, O)$ and $4$ consonants $(D, C, T, N)$.
First,arrange the $5$ vowels in the given order: $E, U, A, I, O$. This can be done in $1$ way.
To satisfy the condition that no two consonants are next to each other,we place the consonants in the spaces created by the vowels. The arrangement of vowels creates $6$ possible spaces (including the ends):
$ \_ V \_ 1 \_  V \_ 2   \_  V \_ 3  \_  V \_ 4  \_  V \_ 5 \_ $
We need to choose $4$ spaces out of these $6$ available spaces to place the $4$ consonants.
Since the consonants must appear in the fixed order $(D, C, T, N)$,there is only $1$ way to arrange them once the $4$ spaces are chosen.
The number of ways to choose $4$ spaces out of $6$ is given by the combination formula $\binom{6}{4}$.
$\binom{6}{4} = \binom{6}{2} = \frac{6 	imes 5}{2 	imes 1} = 15$.
Therefore,there are $15$ such arrangements.

How many ways are there to arrange the letters of the word $EDUCATION$ so that all the following three conditions hold?
- The vowels occur in the same order $(E, U, A, I, O)$.
- The consonants occur in the same order $(D, C, T, N)$.
- No two consonants are next to each other.

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How many ways are there to arrange the letters of the word $EDUCATION$ so that all the following three conditions hold? - The vowels occur in the same order $(E, U, A, I, O)$. - The consonants occur in the same order $(D, C, T, N)$. - No two consonants are next to each other.