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

​\( =210 \times 120 \)​

​\( =25200 \)​