Question

Prove that the number of words which can be formed out of the letters $a,b,c,d,e,f$ taken 3 together, each word containing one vowel at least is 96.

Answer 1

Reqd. no. of words
$={(}^2{C}_1{\times }^4{C}_2{+}^2{C}_2{\times }^4{C}_1)3!$
$=(12+4)\times 6=96$.
Alt. 6 letters, 2 vowels, 4 consonants
Total - No vowel ${=}^6{P}_3{-}^4{C}_3$
$=120-24=96$.