The total number of words that can be formed | vedclass

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(C) The given letters are $\{a, b, c, d, e, f\}$,where vowels are $\{a, e\}$ ($2$ letters) and consonants are $\{b, c, d, f\}$ ($4$ letters). We need

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$96$

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(C) The given letters are $\{a, b, c, d, e, f\}$,where vowels are $\{a, e\}$ ($2$ letters) and consonants are $\{b, c, d, f\}$ ($4$ letters). We need to form $3$-letter words containing at least one vowel. Total words with at least one vowel = (Total words of $3$ letters) - (Words with no vowels). Total words of $3$ letters from $6$ distinct letters = $^6P_3 = 6 	imes 5 	imes 4 = 120$. Words with no vowels (using only consonants $\{b, c, d, f\}$) = $^4P_3 = 4 	imes 3 	imes 2 = 24$. Therefore,the number of words with at least one vowel = $120 - 24 = 96$.

The total number of words that can be formed using the letters $a, b, c, d, e, f$ taken $3$ at a time such that each word contains at least one vowel is:

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