The number of arrangements that can be formed from the letters $a, b, c, d, e, f$ taken 3 at a time without…
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
96
128
24
72
Solution
There are 2 vowels and 4 consonants in the letters $a, b, c, d, e, f$.
If we select one vowel, then number of arrangements
$
={ }^2 C_1 \times{ }^4 C_2 \times 3 !=2 \times \frac{4 \times 3}{2} \times 3 \times 2=72
$
If we select two vowels, then number of arrangements
$
={ }^2 C_2 \times{ }^4 C_1 \times 3 !=1 \times 4 \times 6=24
$
Hence, total number of arrangements
$
=72+24=96
$