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

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

Tardigrade Admin · Accepted Answer

Total number of permutations =(9 !/2 !) Number of those containing 'HIN' =7 ! Number of those containing 'DUS' =(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 =(9 !/2 !)-(7 !+7 !+(7 !/2))+3 × 5 !-3 !=169194

Q. The number of permutations of the letters of the word HINDUSTAN such that neither the pattern ′HIN′ nor ′DUS′ nor ′TAN′ appears, are

166674

169194

166680

181434

Q. The number of permutations of the letters of the word $H I N D U ST A N$ such that neither the pattern $^{'} H I N^{'}$ nor $^{'} D U S^{'}$ nor $^{'} T A N^{'}$ appears, are