The number of arrangements that can be formed | vedclass

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(A) The set of letters is $\{a, b, c, d, e, f\}$,which contains $2$ vowels $(\{a, e\})$ and $4$ consonants $(\{b, c, d, f\})$.We need to form arrangem

Vedclass · Accepted Answer

$96$

Vedclass · Answer

(A) The set of letters is $\{a, b, c, d, e, f\}$,which contains $2$ vowels $(\{a, e\})$ and $4$ consonants $(\{b, c, d, f\})$.We need to form arrangements of $3$ letters containing at least one vowel.Total arrangements with at least one vowel = (Total arrangements of $3$ letters) - (Arrangements with no vowels).Total arrangements of $3$ letters from $6$ distinct letters = $^6P_3 = 6 	imes 5 	imes 4 = 120$.Arrangements with no vowels (only consonants) = $^4P_3 = 4 	imes 3 	imes 2 = 24$.Therefore,the number of arrangements with at least one vowel = $120 - 24 = 96$.

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

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