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

Solution :
Total number of permutations $$ = \frac{{9!}}{{2!}}$$
Number of those containing $$'HIN' = 7!$$
Number of those containing $$'DUS' = \frac{{7!}}{{2!}}$$
Number of those containing $$'TAN' = 7!$$
Number of those containing $$'HIN'$$  and $$'DUS' = 5!$$
Number of those containing $$'HIN'$$  and $$'TAN' = 5!$$
Number of those containing $$'TAN'$$  and $$'DUS' = 5!$$
Number of those containing $$'HIN', 'DUS'$$   and $$'TAN' = 3!$$
Required number
$$ = \frac{{9!}}{{2!}}\left( {7!\, + 7!\, + \frac{{7!}}{2}} \right) + 3 \times 5!\, - 3! = 169194.$$