Question

The number of permutations of the letters of the word HINDUSTAN such that neither the pattern 'HIN' nor 'DUS' nor 'TAN' appears are

• A. $166674$
• B. $169194$
• C. $166680$
• D. $181434$

Answer 1

The correct option is B

$169194$

Explanation for the correct option:

Find the required number of permutations.

The word ‘HINDUSTAN’ is given.

The total number of permutations of the given word is $\frac{9!}{2!}$.

The total number of permutations in which ‘$HIN$’ appears as a block is $7!$.

The total number of permutations in which ‘$DUS$’ appears as a block is $7!$.

The total number of permutations in which ‘$TAN$’ appears as a block is $\frac{7!}{2!}$.

The total number of permutations in which ‘$HIN$’ and ‘$DUS$’ appears as a block is $5!$. This also includes ‘HINDUSTAN’.

The total number of permutations in which ‘$DUS$’ and ‘$TAN$’ appears as a block is $5!$. This also includes ‘HINDUSTAN’.

The total number of permutations in which ‘$HIN$’ and ‘$TAN$’ appears as a block is $5!$. This also includes ‘HINDUSTAN’.

So, the total number of required permutations can be given by $\frac{9!}{2!}-\left(7!+7!+\frac{7!}{2!}-3(5!)+3!\right)$.

Therefore, The number of permutations of the letters of the word HINDUSTAN such that neither the pattern '$HIN$' nor '$DUS$' nor '$TAN$' appears are $169194$.

Hence, option $B$ is the correct answer.