Question

From $3$ capitals, $5$ consonants, and $4$ vowels, how many words can be made, each containing $3$ consonants and $2$ vowels, and beginning with a capital?

Answer 1

Number of ways of choosing $3$ consonants from $5$ consonants $=$ ${}^5{C}_3=10\text{ways.}$

Number of ways of choosing $2$ vowels from $4$ vowels $=$ ${}^4{C}_2=6\text{ways.}$

Number of ways of choosing $1$ capitals from $3$ capitals $=$ ${}^3{C}_1=3\text{ways.}$

This would result in $6$ lettered word with $1^{\text{st}}$ letter as capital and remaining $5$ letters can be anything out of vowels and consonants.

Hence, the first position is fixed by capitals and in remaining $5$ places $5$ letters can be arranged in ${}^5{P}_5=5!=120\text{ways.}$

Hence, total number of ways $=$ $10\times 6\times 3\times 120=21600\text{ways.}$