Question

Out of $7$ consonants and $4$ vowels, how many words can be made each containing $3$ consonants and $2$ vowels?

Answer 1

The number of ways of choosing three consonants is ${}^7{C}_3$7C3, and the number of ways of choosing $2$ vowels is ${}^4{C}_2$; and since each of the first groups can be associated with each of the second, the number of combined groups, each containing $3$ consonants and $2$ vowels, is ${}^7{C}_3\times {}^4{C}_2$.

Further, each of these groups contains $5$ letters, which may be arranged among themselves in $5!$ ways. Hence the required number of words $={}^7{C}_3\times {}^4{C}_2\times 5!=25200$.