I borrowed a dictionary from Ashish!

Looking backwards.
Lets arrange letters of BRILLIANT is descending alphabetical order. It is TRNLLIIBA.

$\text{First letter}$
First lets see words starting with T. Its is the same as in T NAILLIRB. So, we proceed to the next letter.

$\text{Second letter}$
First in reverse alphabetical order comes R, but we need N. So, there would be $\dfrac{7!}{2! × 2!} = 1260$ such words. Then comes N which is what we want(T N AILLIRB). So, lets proceed to the next letter.

$\text{Third letter}$
First comes R but we need A, so there are $\dfrac{6!}{2! × 2!} = 180$ such words. Next comes L but we need A, so there are $\dfrac{6!}{2!} = 360$ such words. Next comes I but we need A, so there are $\dfrac{6!}{2!} = 360$ such words. Next comes B but we need A, so there are $\dfrac{6!}{2! × 2!} = 180$ such words. Next comes A which we need (TN A ILLIRB). So, we proceed to the next letter.

$\text{Fourth letter}$
First comes R but we need I, so there are $\dfrac{5!}{2! × 2!} = 30$ such words. Next comes L but we need I, so there are $\dfrac{5!}{2!} = 60$ such words. Next comes I which we want (TNA I LLIRB). So, we proceed to the next letter.

$\text{Fifth letter}$
First comes R but we need L, so there are $\dfrac{4!}{2!} = 12$ such words. Next comes L which we want (TNAI L LIRB). So, we proceed to the next letter.

$\text{Sixth letter}$
First comes R but we need L, so there are $3! = 6$ such words. Next comes L which we want (TNAIL L IRB). So, we proceed to the next letter.

$\text{Seventh letter}$
First comes R but we need I, so there are $2! = 2$ such words. Next comes I which we want (TNAILL I RB). So, we proceed to the next letter.

$\text{Eight letter}$
First comes R which we want (TNAILLI R B). So, we proceed to the next letter.

$\text{Ninth letter}$
First comes B which we want (TNAILLIR B ).

So, the word is the $1260 + 180 + 360 + 360 + 180 + 30 + 60 + 12 + 6 + 2 = 2450$ th letter from the back of the dictionary.

Now, the total number of pages in the dictionary would be the total number of letter that can be formed from BRILLIANT, which is $\dfrac{9!}{2! × 2!} = 90720$ .

So, the page in which TNAILLIRB would be present = $90720 - 2450 = \boxed{88270}$ .

$\boxed{\text{Q.E.D}}$

Now, before the community gives me the page number, I shall try to count the pages myself first.

Before we begin counting, let's list out all the words before "TNAILLIRB".

First, the dictionary begins with words that start with A.

After that, we go through words that start with B, I, L, N and R before reaching T.

Then, the word starts with TN, so we go through words beginning with TA, TB, TI and TL.

The 3rd letter is A, which comes first.

The 4th letter is I, so we go through words beginning with TNAB.

The 5th letter is L, so we go through words beginning with TNAIB, TNAII

The 6th letter is L, so we go through words beginning with TNAILB, TNAILI

The 7th letter is I, so we go through words beginning with TNAILLB

After that, we are left with TNAILLIBR and TNAILLIRB, which are 2 additional words.

Therefore, we want to find the sum of all these words.

Now, to count all the words here:

Words that start with A (there are 2 I and 2 L) $=\dfrac{8!}{2! \times 2!} = 10080$

Words that start with B (there are 2 I and 2 L) $=\dfrac{8!}{2! \times 2!} = 10080$

Words that start with I (there are 2 L) $=\dfrac{8!}{2!} = 20160$

Words that start with L (there are 2 I) $=\dfrac{8!}{2!} = 20160$

Words that start with N (there are 2 I and 2 L) $=\dfrac{8!}{2! \times 2!} = 10080$

Words that start with R (there are 2 I and 2 L) $=\dfrac{8!}{2! \times 2!} = 10080$

Words that start with TA (there are 2 I and 2 L) $=\dfrac{7!}{2! \times 2!} = 1260$

Words that start with TB (there are 2 I and 2 L) $=\dfrac{7!}{2! \times 2!} = 1260$

Words that start with TI (there are 2 L) $=\dfrac{7!}{2!} = 2520$

Words that start with TL (there are 2 I) $=\dfrac{7!}{2!} = 2520$

Words that start with TNAB (there are 2 I and 2 L) $=\dfrac{5!}{2! \times 2!} = 30$

Words that start with TNAIB (there are 2 L) $=\dfrac{4!}{2!} = 12$

Words that start with TNAII (there are 2 L) $=\dfrac{4!}{2!} = 12$

Words that start with TNAILB $=3! = 6$

Words that start with TNAILI $=3! = 6$

Words that start with TNAILLB $=2!=2$

TNAILLIBR - $1$ word

TNAILLIRB - $1$ word

The page number = Total number of words

$=10080 +10080 +20160+20160+10080+10080+1260+1260+2520+2520+30+12+12+6+6+2+1+1\\ =\boxed{88270}$

Alternatively, you can begin counting from the last page. I'll let someone else write out that solution.