Question

Find the number of seven lettered words each consisting of three vowels and four consonants which can be formed using the letters of the word ${}^{\prime }DIFFERENTIATIO{N}^{\prime }$.

• A. $432551$
• B. $325710$
• C. $252000$
• D. $644004$

Answer 1

The correct option is B $325710$

DIFFERENTIATION

Vowels - I,E,E,I,A,I,O

Consonants - D,F,F,R,N,N,T,T

No of ways of selecting :

3 vowels

$⟶\text{All different}={}^4{C}_3=4⟶\text{2 same, 1 different}={}^2{C}_1{}^3{C}_2=6⟶\text{3 same}=1$

4 consonants

$⟶\text{All different}={}^5{C}_4=5⟶\text{2 same, 2 different}={}^3{C}_1\times {}^4{C}_2=18⟶\text{2 pair same}={}^3{C}_2=3$

$\therefore$ Required number of ways

$=4\times 5\times 7!+4\times 18\times \frac{7!}{2!}+4\times 3\times \frac{7!}{2!2!}+6\times 5\times \frac{7!}{2!}+6\times 18\times \frac{7!}{2!2!}+6\times 3\times \frac{7!}{2!2!2!}+1\times 5\times \frac{7!}{3!}+1\times 18\times \frac{7!}{3!2!}+1\times 3\times \frac{7!}{2!2!3!}=325710$