Question:
The English alphabet has 21 consonants and 5 vowels. How many words with two different consonants and three different vowels can be formed from the alphabet?
Solution:
2 consonants out of 21 consonants can be chosen in ${ }^{21} C_{2}$ ways. 3 vowels out of 5 vowels can be chosen in ${ }^{5} C_{3}$ ways. Length of the word is $=(2+3)=5$ And also 5
letters can be written in 5! Ways. Therefore, the number of words can be formed is = $\left({ }^{21} \mathrm{C}_{2} \times{ }^{5} \mathrm{C}_{3} \times 5 !\right)=252000 .$
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