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

Question:

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

Solution:

2 out of 5 vowels and 3 out of 17 consonants can be chosen in ${ }^{5} C_{2} \times{ }^{17} C_{3}$ ways.

Thus, there are ${ }^{5} C_{2} \times{ }^{17} C_{3}$ groups, each containing 2 vowels and 3 consonants.

Each group contains 5 letters, which can be arranged in $5 !$ ways.

$\therefore$ Required number of words $=\left({ }^{5} C_{2} \times{ }^{17} C_{3}\right) 5 !=6800 \times 120=816000$

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