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Q.
The number of permutations of the letters of the word $HINDUSTAN$ such that neither the pattern $'HIN'$ nor $'DUS'$ nor $'TAN'$ appears, are
Permutations and Combinations
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
$=\frac{9 !}{2 !}-\left(7 !+7 !+\frac{7 !}{2}\right)+3 \times 5 !-3 !=169194$