Rank of a word

In work SILENT, there are six distinct letters. Therefore, rearranging the letters, there are 6! words possible.
From these 6! words, we need to find the rank lo LISTEN if all the words are arranged in alphabetical order.
To solve such questions, first we need to write letters of the given words in alphabetical order, i.e. E, I, L, N, S and T.
E is first letter(E, _, _, _, _, _), number of words possible = 5! = 120
I is first letter(I, _, _, _,_, _), number of words possible = 5! = 120
L, E, _, _, _, _ -> 4!
L, I, E, _, _, _ -> 3!
L, I, N, _, _, _ -> 3!
L, I, S, E, _, _ -> 2!
L, I, S, N, _, _ -> 2!
L, I, S, T, E, N -> 1
Rank = 120 + 120 + 24 + 6 + 6 + 2 + 2 + 1 = 281

Arranging the letters of the word STUDENT, we get 2520 7-letter words possible. In this question, we need to find the rank of word STUDENT from all 2520 words. 

Alphabetical order - D,E,N,S,T,T,U

D, _, _, _, _, _, _ ---> $$\frac{6!}{2!}$$ words

E, _, _, _, _, _, _ ---> $$\frac{6!}{2!}$$ words

N, _, _, _, _, _, _ ---> $$\frac{6!}{2!}$$ words

S, D, _, _, _, _, _ ---> $$\frac{5!}{2!}$$ words

S, E, _, _, _, _, _ ---> $$\frac{5!}{2!}$$ words

S, N, _, _, _, _, _ ---> $$\frac{5!}{2!}$$ words

S, T, D, _, _, _, _ ---> $$4!$$ words

S, T, E, _, _, _, _ ---> $$4!$$ words

S, T, N, _, _, _, _ ---> $$4!$$ words

S, T, T, _, _, _, _ ---> $$4!$$ words

S, T, U, D, E, N, T ---> 1

Rank = $$\frac{3\cdot6!}{2!}+\ \frac{\ 3\cdot5!}{2!}+4\cdot4! + 1$$ = 1357

The answer is option D.

Arranging EXEMPLARY in alphabetical order we get AEELMPRXY. The first words of the 7 letter list would start from A.

For A, removing the extra E, $$^7 P _6$$ words can be formed from the distinct letters. Now let us take the case when both E are present in the mix. So, out of the 7 letters, 3 letters are A, and 2 Es. So, the other 4 letters can be selected as $$^6C_4$$. So, the words can be arranged in $$^6 C _4 \times (6!/2!)$$ ways. Hence there are $$^7 P _6 + ^6 C _4*(6!/2!)$$ 7-letter words starting with A.
Next in line would be words starting with E.
Let us first consider words of form, EA _ _ _ _ _.
The 5 digits can be arranged in $$^7P_5 ways$$.

Same number of words will be there of the form, EE _ _ _ _ _, EL _ _ _ _ _, EM _ _ _ _ _, EP _ _ _ _ _ , and ER _ _ _ _ _.

Each of these letters produces $$^7 P _5$$ words.
By this way we get that rank= $$(^7 P _6 + ^6 C _4 * (6!/2!))+(6* ^7 P _5) +(2 * ^5 P _3)+(2* ^4 P _2) + (1* ^3 P _1) +1 = 25708$$.