The total number of words that can be formed | vedclass

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(C) The given letters are ${a, b, c, d, e, f}$. There are $2$ vowels ${a, e}$ and $4$ consonants ${b, c, d, f}$.We need to form $3$-letter words conta

Vedclass · Accepted Answer

$96$

Vedclass · Answer

(C) The given letters are ${a, b, c, d, e, f}$. There are $2$ vowels ${a, e}$ and $4$ consonants ${b, c, d, f}$.We need to form $3$-letter words containing at least one vowel.Case $1$: Words with exactly $1$ vowel and $2$ consonants.Number of ways to select $1$ vowel from $2$ and $2$ consonants from $4$ is $^2C_1 	imes ^4C_2 = 2 	imes 6 = 12$.These $3$ letters can be arranged in $3! = 6$ ways.Total words = $12 	imes 6 = 72$.Case $2$: Words with exactly $2$ vowels and $1$ consonant.Number of ways to select $2$ vowels from $2$ and $1$ consonant from $4$ is $^2C_2 	imes ^4C_1 = 1 	imes 4 = 4$.These $3$ letters can be arranged in $3! = 6$ ways.Total words = $4 	imes 6 = 24$.Total number of words = $72 + 24 = 96$.

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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