# The English alphabet has 5 vowels and 21 consonants.How many words with two different vowels and two different consonants can be formed from the alphabet?

$\begin{array}{1 1}(A)\;50000\\(B)\;50100\\(C)\;50300\\(D)\;50400\end{array}$

Toolbox:
• $C(n,r)=\large\frac{n!}{r!(n-r)!}$
• $n!=n(n-1)(n-2)(n-3).......(3)(2)(1)$
2 vowels can be chosen in $5C_2$ ways
2 consonants can be chosen in $21C_2$ ways
4 letters can be arranged in 4! ways
$\therefore$ The number of words consisting of 2 vowels and 2 consonants
$\Rightarrow 5C_2\times {21}C_2\times 4!$
$\Rightarrow \large\frac{5\times 4}{1\times 2}\times \frac{21\times 20}{1\times 2}$$\times 24$
$\Rightarrow 10\times 210\times 24$
$\Rightarrow 50400$
Hence (D) is the correct answer.