Question

The number of different words that can be formed out of the letters of the word 'MORADABAD' taken four at a time is?

• A. $500$
• B. $600$
• C. $620$
• D. $626$

Answer 1

The correct option is D

$626$

Explanation for the correct option:

Find the number of four letters words.

The word 'MORADABAD' is given.

Since there are $9$ letters in total out of which $3$ are $A\text{'}s$ and $2$ are $D\text{'}s$.

Now, the number of words formed by all different letters is given by ${}^{6}C_4·4!=360$.

The number of words formed by $2A\text{'}s$ and $2D\text{'}s$ is given by $\frac{4!}{2!·2!}=6$.

The number of words formed by $2A\text{'}s$ and two different letters is given by $\frac{{}^{5}C_2}{2!}·4!=120$.

The number of words formed by $3A\text{'}s$ and one different letter is given by $\frac{{}^{3}C_3}{3!}·{}^{5}C_1·4!=20$.

The number of words formed by $2D\text{'}s$ and two different letters is given by $\frac{{}^{5}C_2}{2!}·4!=120$.

Therefore, the total number of four letters words formed out of the letters of the word ‘MORADABAD’ is given by $360+6+120+20+120=626$.

Hence, option $D$ is the correct answer.