Question

What is the number of permutations of the letters of the word HINDUSTAN such that neither the pattern
"HIN" nor "DUS" nor "TAN" appears ?

Answer 1

What is the number of permutations of the letters of the word HINDUSTAN such that neither the pattern "HIN" nor "DUS" nor "TAN" appears?

Solution

Number of permutations of the letters of the word which has n letters is n!

If some of letters is doubled Number of permutations is $\frac{n!}{2}$, 2 letters are doubled

Number of permutations is $\frac{n!}{2 \times 2}$ ...

If we have fixed block than we work with it like as letter:

'HIN' 'D' 'U' 'S' 'T' 'A' 'N' - 7 letters, Number of permutations 7!

(a) Total number of permutations $= \frac{9!}{2}$, since N is repeated.

(b) Number of permutations in which HIN comes as a block = 7!

Number of permutations in which TAN comes as a block = 7!

Number of permutations in which DUS comes as a block $= \frac{7!}{2}$.

(c) This includes both HIN and TAN comes as blocks = 5!

same is true for the other two pairs.

(d) Number of permutations in which half three blocks come = 3!

$\therefore$ required number of permutations

$$\begin{array}{l} = (a) - \{(b) - (c) + (d)\} = \frac{9!}{2} - \left\{7! + 7! + \frac{7!}{2} + 3(5!) + 3!\right\} = \\ = \frac{362880}{2} - \left\{5040 + 5040 + \frac{5040}{2} - 3 \times 120 + 6\right\} = 169194 \\ \end{array}$$

Answer: 169194.