Question

How many different words, each containing $2$ vowels and $3$ consonants can be formed with $5$ vowels and $17$ consonants ?

Answer 1

Given: Total vowels are $5$ and total consonants are $17$.
The required different words that contain $2$ vowels and $3$ consonants:
The number of ways to select $2$ vowels out of $5$ vowels are ${}^5{C}_2$ ways.
Similarly, the number of ways to select $3$ consonants out of $17$ consonants are ${}^{17}{C}_3$ ways.
Thus, the total selection ${=}^5{C}_2{\times }^{17}{C}_3$.

Now, the $5$ letters in each selection can be arranged in $5!$ ways.

Therefore, the total number of words ${=}^5{C}_2{\times }^{17}{C}_3\times 5!$

$=\frac{5!}{2!3!}\times \frac{17!}{3!\times 14!}\times 120$

$=\frac{5\times 4}{2}\times \frac{17\times 16\times 15}{3\times 2}\times 120$

$=10\times 17\times 8\times 5\times 120$

$=400\times 17\times 120$

$=6800\times 120=816000$