Question

The letters of the word $INCIDENT$ are permuted and all the permutations are arranged in an alphabetical order as in English dictionary. Let $m$ be the rank of the word $INCIDENT$ in the dictionary. Then the value of $\left[\frac{m}{100}\right]$ is

Here, $\left[x\right]$ denotes the greatest integer less than or equal to $x$.

Answer 1

The letters of the word '$\text{INCIDENT}$' are :
$\text{C,D,E,I,I,N,N,T}$

Number of words beginning with $\text{C,D,E}=3\times \frac{7!}{2!\cdot 2!}=3780$

Number of words beginning with $\text{IC,ID,IE,II}=4\times \frac{6!}{2!}=1440$

Number of words beginning with $\text{INCD,INCE}=2\times 4!=48$

Therefore, rank of $\text{INCIDENT,}m=3780+1440+48+1=5269$

Hence, $\left[\frac{5269}{100}\right]=52$