The English alphabet has $5$ vowels and $21$ consonants. How many words with two different vowels and $2$ different consonants can be formed from the alphabet?
The English alphabet has $5$ vowels and $21$ consonants. How many words with two different vowels and $2$ different consonants can be formed from the alphabet?
$2$ different vowels and $2$ different consonants are to be selected from the English alphabet. since there are $5$ vowels in the English alphabet, number of ways of selecting $2$ different vowels from the alphabet $=\,^{5} C_{2}=\frac{5 !}{2 ! 3 !}=10$
since there are $21$ consonants in the English alphabet, number of ways of selecting $2$ different consonants from the alphabet $=\,^{21} C_{2}=\frac{21 !}{2119 !}=210$
Therefore, number of combinations of $2$ different vowels and $2$ different consonants $=10 \times 210=2100$
Each of these $2100 $ combinations has $4$ letters, which can be arranged among themselves in $4 !$ ways.
Therefore, required number of words $=2100 \times 4 !=50400$
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