Question

Out of $8$ consonants and $3$ vowels, how many set of letters of $3$ consonants and $2$ vowels can be formed?

• A. $168$
• B. $120$
• C. $20160$
• D. $288$

Answer 1

The correct option is C $20160$

Number of ways of selecting $3$ consonants from $8$ consonats $={}^8{C}_3$

Number of ways of selecting $2$ vowels from $3$ vowels $={}^3{C}_2$

Number of ways of selecting $3$ consonants from $8$ consonants and $2$ vowels from $3$ vowels would be obtained by multiplying ${}^8{C}_3$ by ${}^3{C}_2$:
$={}^8{C}_3\times {}^3{C}_2$
$=\left(\frac{8!}{3!(8-3)!}\right)\times \left(\frac{3!}{2!(3-2)!}\right)$
$=\frac{8\times 7\times 6}{3\times 2\times 1}\times \frac{3\times 2\times 1}{2\times 1}$
$=56\times 3$
$=168$

This means that we have $168$ groups where each group contains a total of $5$ letters ($3$ consonants and $2$ vowels).

For example, if $FENIL$ is a set of letters then $NELIF$ is another set of letters. This means that we can arrange $5$ letters among themselves.

Number of ways of arranging $5$ letters among themselves $=5!=5\times 4\times 3\times 2\times 1$
$=120$

Number of set of letters that can be formed by selecting $3$ consonants from $8$ consonants and $2$ vowels from $3$ vowels $=168\times 120$
$=20160$


Therefore, option (c.) is the correct answer.