Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

Question:

Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

A. 25200               B. 21300

C. 24400               D. 210


Answer:

3 consonants can be selected from 7 consonants in $^7C_3$ ways.

2 vowels can be selected from 4 vowels in $^4C_2$ ways.

$\therefore$ by multiplication principle, 
the number of selecting 3 consonants and 2 vowels is
$=^7C_3 \times ^4C_2$ 

$=\frac{7!}{3!4!} \times \frac{4!}{2!2!}$ 

$=\frac{7.6.5}{3.2.1} \times \frac{4.3}{2.1}$ 

$=35 \times 6$ 

$=210$ 

Now, the number of ways of arranging 5 letters among themselves 
$=5!$
$=120$ 

$\therefore$ the total number of words of 3 consonants and 2 vowels
$=210 \times 120$ 
$=25200$ 

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