Question

The English alphabet has $5$ vowel and $21$ consonants how many words with two different vowels and two different consonants can be formed from the alphabet?

Answer 1

Finding number of ways of selecting $2$ vowels and $2$ consonants.

Number of vowels $=5$

Number of vowels to be selected $=2$

Number of consonants $=21$

Number of consonants to be selected $=2$

$\therefore$ Number of ways selecting $2$ vowels and $2$ consonants $={}^5{C}_2\times {}^{21}{C}_2$

$=\frac{5!}{2!\times 3!}\times \frac{21!}{(21-2)!2!}$

$=\frac{5\times 4\times 3!}{2!\times 3!}\times \frac{21\times 20\times 19!}{19!\times 2!}$

$=\frac{5\times 4}{2\times 1}\times \frac{21\times 20}{2\times 1}=2100$

Step $2$: Finding number of arrangements of $4$ letter.

Number of arrangements of $4$ letters

$={}^4{P}_4$

$=\frac{4!}{(4-4)!}$

$=\frac{4\times 3\times 2\times 1}{0!}=24$

Step $3$: Finding total number of ways.

$\therefore$ Total number of words $=2100\times 24$

$=50400$