The no. of permutations of the letters of the word HINDUSTANsuch that neither the pattern 'HIN' nor 'DUS' nor 'TAN' appears are
1. 166674
2.169194
3.166680
4.181434

total number of letters = 9, in which N is repeated twice.
therefore total number of permutations = 9! / 2
the number of permutations in which HIN comes as block = 7!  [9-3+1=7
number of permutations in which DUS comes as block = 7!
number of permutations in which TAN comes as block = 7! / 2
the number of permutations in which HIN and DUS comes as block = 5!
same is true for other two pairs.
number of permutations in which all the three blocks come = 3!
therefore the required number of permutations
=9!2-(7!+7!+7!2-3*5!+3!)=3628802-5040+5040+50402-3*120+6=181440-10080+2520-360+6=181440-12246=169194
thus option (2) is correct
hope this helps you

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