Kacey is handing out candy for Halloween. She has only $15$ candies left when a ghost, a goblin, and a vampire arrive at her door. She wants to give each trick-or-treater at least one candy, but she does not want to give any two the same number of candies. How many ways can she distribute all $15$ identical candies to the three trick-or-treaters given these restrictions?

$\text{(A) }91\qquad\text{(B) }90\qquad\text{(C) }81\qquad\text{(D) }72\qquad\text{(E) }70$
1. **Initial Setup**: We start by giving each of the three trick-or-treaters at least one candy. This ensures that each one gets at least one candy, leaving us with \(15 - 3 = 12\) candies to distribute.

2. **Stars and Bars**: We use the stars and bars method to find the number of ways to distribute 12 candies among 3 trick-or-treaters without any restrictions. The formula for distributing \(n\) identical items into \(k\) distinct groups is given by:
   \[
   \binom{n+k-1}{k-1}
   \]
   Here, \(n = 12\) and \(k = 3\), so we have:
   \[
   \binom{12+3-1}{3-1} = \binom{14}{2} = 91
   \]

3. **Subtracting Invalid Cases**: We need to subtract the cases where two or more trick-or-treaters receive the same number of candies. We will use casework to count these invalid distributions.

   - **Case 1**: Two trick-or-treaters receive the same number of candies.
     - \(0-0-12\): There are 3 ways to distribute (choosing which two get 0 candies).
     - \(1-1-10\): There are 3 ways to distribute.
     - \(2-2-8\): There are 3 ways to distribute.
     - \(3-3-6\): There are 3 ways to distribute.
     - \(4-4-4\): There is 1 way to distribute.
     - \(5-5-2\): There are 3 ways to distribute.
     - \(6-6-0\): There are 3 ways to distribute.

     Summing these, we get:
     \[
     3 + 3 + 3 + 3 + 1 + 3 + 3 = 19
     \]

4. **Final Calculation**: Subtract the invalid cases from the total number of unrestricted distributions:
   \[
   91 - 19 = 72
   \]

Thus, the number of ways Kacey can distribute the 15 candies to the three trick-or-treaters, ensuring each gets at least one candy and no two get the same number of candies, is:

\(\boxed{72}\)