The number of permutations of the letters of the word HINDUSTAN such that neither the pattern 'H I N.nor 'D U S' nor 'T A N' appears are

166680

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181434

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166674

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169194

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Solution

The correct option is D 169194
The word "HINDUSTAN" has a letters in which the letter
n is repeated twice.
The number of permutations of the word "HINDUSTAN"
$= \frac{9!}{2!}$ [since n is repeated twice ]
The number of permutations in which the pattern "HIN" appeases
=7!
The numbers of permutations in which the pattern "DUS"appeases.
$= \frac{7!}{2!}$ [since N is separated twice]
The numbers of permutations in which the pattern "TAN"appease
=7!
The number of permutations in which "HIN"and "DUS"appease
=5!
The numbers of permutations in which "DUS"and "TAN"appease
=5!
The numbers of permutations in which "Hin" and "TAN" appease
=6!
The numbers of permutations in which "HIN","DUS" and "TAN"
appears = 3!
Hence, the numbers of permutations of the word "HINDUSTAN"
such that neither the pattern "HIN" nor "DUS" nor "TAN"
appease= $\frac{9!}{2!} - 7! - \frac{7!}{2!} - 7! + 5! + 5! + 5! - 3!$
=181440-5040-2520-5040-120+120+120-6
=169194

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