The number of words (with or without meaning) | vedclass

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Question

(C) The word $LETTER$ contains $6$ letters: $L, E, T, T, E, R$.The vowels are $E, E$ and the consonants are $L, T, T, R$.First,arrange the consonants

Vedclass · Accepted Answer

$120$

Vedclass · Answer

(C) The word $LETTER$ contains $6$ letters: $L, E, T, T, E, R$.The vowels are $E, E$ and the consonants are $L, T, T, R$.First,arrange the consonants $L, T, T, R$. The number of ways to arrange these is $\frac{4!}{2!} = \frac{24}{2} = 12$.There are $5$ possible gaps created by these $4$ consonants (including the ends) where the $2$ vowels can be placed: $\_ L \_ T \_ T \_ R \_$.The number of ways to choose $2$ gaps out of $5$ is $^{5}C_{2} = 10$.The $2$ vowels $(E, E)$ are identical,so they can be arranged in the chosen gaps in $\frac{2!}{2!} = 1$ way.Total number of words = $12 	imes 10 	imes 1 = 120$.

The number of words (with or without meaning) that can be formed from all the letters of the word $LETTER$ in which vowels never come together is

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