Question

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

• A. $96$
• B. $128$
• C. $24$
• D. $72$

Answer 1

Answer: (A)

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
$=^{2}C_{1} \times^{4}C_{2}\times3!=2\times \frac{4\times3}{2}\times3\times2=72$
If we select two vowels, then number of arrangements
$=^{2}C_{2}\times^{4}C_{1}\times3!=1\times4\times6=24$
Hence, total number of arrangements
$= 72 + 24 = 96$