The number of 4-letter words,with or without | vedclass

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Question

(C) The word $UNIVERSE$ contains $8$ letters: $U, N, I, V, E, R, S, E$.Distinct letters are $U, N, I, V, E, R, S$.There are $3$ vowels: $\{U, I, E\}$

Vedclass · Accepted Answer

$432$

Vedclass · Answer

(C) The word $UNIVERSE$ contains $8$ letters: $U, N, I, V, E, R, S, E$.Distinct letters are $U, N, I, V, E, R, S$.There are $3$ vowels: $\{U, I, E\}$ and $4$ consonants: $\{N, V, R, S\}$.We need to select $2$ vowels out of $3$ and $2$ consonants out of $4$.Number of ways to select the letters = $\binom{3}{2} 	imes \binom{4}{2} = 3 	imes 6 = 18$.Each selection consists of $4$ distinct letters,which can be arranged in $4! = 24$ ways.Total number of words = $18 	imes 24 = 432$.

The number of $4$-letter words,with or without meaning,each consisting of $2$ vowels and $2$ consonants,which can be formed from the letters of the word $UNIVERSE$ without repetition is $.........$.

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