The number of arrangements that can be formed | vedclass

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(A) The set of letters is ${a, b, c, d, e, f}$. There are $2$ vowels $({a, e})$ and $4$ consonants $({b, c, d, f})$.We need to form arrangements of $3

Vedclass · Accepted Answer

$96$

Vedclass · Answer

(A) The set of letters is ${a, b, c, d, e, f}$. There are $2$ vowels $({a, e})$ and $4$ consonants $({b, c, d, f})$.We need to form arrangements of $3$ letters such that at least one vowel is included.Case $1$: Exactly one vowel is selected.Number of ways to select $1$ vowel and $2$ consonants is $^2C_1 	imes ^4C_2 = 2 	imes 6 = 12$.Number of arrangements for these selections is $12 	imes 3! = 12 	imes 6 = 72$.Case $2$: Exactly two vowels are selected.Number of ways to select $2$ vowels and $1$ consonant is $^2C_2 	imes ^4C_1 = 1 	imes 4 = 4$.Number of arrangements for these selections is $4 	imes 3! = 4 	imes 6 = 24$.Total number of arrangements $= 72 + 24 = 96$.

The number of arrangements that can be formed from the letters $a, b, c, d, e, f$ taken $3$ at a time without repetition and each arrangement containing at least one vowel,is

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