We need the dictionary rank of the word MOTHER among all permutations of its letters.

The letters are: $M, O, T, H, E, R$

Arrange them in alphabetical order: $E < H < M < O < R < T$

Since all letters are distinct, we count how many valid permutations come before MOTHER in dictionary order.

Step 1: First letter = M

We count words beginning with a letter smaller than $M$.

Smaller letters available: $E, H$

For each such choice, the remaining $5$ letters can be arranged in: $5! = 120$

So count: $2 \times 120 = 240$

Step 2: Fix first letter M, second letter = O

Remaining letters are: $E, H, O, R, T$

Now count words beginning with $M$ and having second letter smaller than $O$.

Smaller available letters than $O$: $E, H$

For each such choice, remaining $4$ letters can be arranged in: $4! = 24$

So count: $2 \times 24 = 48$

Running total: $240 + 48 = 288$

Step 3: Fix MO, third letter = T

Remaining letters are: $E, H, R, T$

Now count words beginning with $MO$ and having third letter smaller than $T$.

Smaller available letters than $T$: $E, H, R$

For each such choice, remaining $3$ letters can be arranged in: $3! = 6$

So count: $3 \times 6 = 18$

Running total: $288 + 18 = 306$

Step 4: Fix MOT, fourth letter = H

Remaining letters are: $E, H, R$

Letters smaller than $H$: only $E$

For this choice, remaining $2$ letters can be arranged in: $2! = 2$

So count: $1 \times 2 = 2$

Running total: $306 + 2 = 308$

Step 5: Fix MOTH, fifth letter = E

Remaining letters are: $E, R$

Letters smaller than $E$: none

So contribution: $0$

Step 6: Fix MOTHE, sixth letter = R

Only one letter remains, so no additional permutations come before it.

Step 7: Add 1 for the word itself

So rank of MOTHER is: $308 + 1 = 309$

Final Answer

The position of the word MOTHER is: $\boxed{309}$

The derived answer matches the stored correct answer.