How many ways are there to insert $+$'s between the digits of $111111111111111$ (fifteen $1$'s) so that the result will be a multiple of $30$?
To solve this problem, we need to determine how many ways we can insert $+$ signs between the digits of $111111111111111$ (fifteen $1$'s) such that the resulting expression is a multiple of $30$. 

1. **Determine the conditions for the number to be a multiple of $30$:**
   - A number is a multiple of $30$ if and only if it is a multiple of both $3$ and $10$.
   - A number is a multiple of $3$ if the sum of its digits is a multiple of $3$. Since all digits are $1$, the sum of the digits is $15$, which is already a multiple of $3$. Therefore, any way of inserting $+$ signs will result in a number that is a multiple of $3$.
   - A number is a multiple of $10$ if and only if it ends in $0$. Since we are only using the digit $1$, the only way to achieve this is to have one of the resulting numbers end in $0$. This is impossible with the given digits. However, we can interpret the problem as needing the sum of the resulting numbers to end in $0$.

2. **Determine the number of splits needed:**
   - To ensure the sum of the resulting numbers ends in $0$, we need to split the sequence into $10$ different numbers. This is because the sum of $10$ ones is $10$, which ends in $0$.

3. **Calculate the number of ways to insert $+$ signs:**
   - We need to insert $9$ $+$ signs among the $14$ possible positions (between the $15$ digits) to create $10$ different numbers.
   - The number of ways to choose $9$ positions out of $14$ is given by the binomial coefficient $\binom{14}{9}$.

4. **Simplify the binomial coefficient:**
   \[
   \binom{14}{9} = \binom{14}{5} = \frac{14!}{5!(14-5)!} = \frac{14!}{5! \cdot 9!}
   \]
   - Calculate the factorials:
     \[
     14! = 87178291200, \quad 5! = 120, \quad 9! = 362880
     \]
   - Substitute and simplify:
     \[
     \binom{14}{5} = \frac{87178291200}{120 \cdot 362880} = \frac{87178291200}{43545600} = 2002
     \]

Conclusion:
\[
\boxed{2002}
\]