In how many ways can $17$ identical red and $10$ identical white balls be distributed into $4$ distinct boxes such that the number of red balls is greater than the number of white balls in each box?

$ 
\textbf{(A)}\ 5462
\qquad\textbf{(B)}\ 5586
\qquad\textbf{(C)}\ 5664
\qquad\textbf{(D)}\ 5720
\qquad\textbf{(E)}\ 5848
$
1. **Glue each white ball to a red ball**: 
   - We start by pairing each of the 10 white balls with a red ball. This results in 10 white-and-red pairs. 
   - We now need to distribute these 10 pairs into 4 distinct boxes. 

2. **Distribute the 10 white-and-red pairs**:
   - The problem of distributing 10 pairs into 4 boxes (with empty boxes allowed) can be solved using the stars and bars method. 
   - The formula for distributing \( n \) identical items into \( k \) distinct boxes is given by:
     \[
     \binom{n+k-1}{k-1}
     \]
   - Here, \( n = 10 \) and \( k = 4 \). Therefore, the number of ways to distribute the 10 pairs is:
     \[
     \binom{10+4-1}{4-1} = \binom{13}{3}
     \]

3. **Distribute the remaining 7 red balls**:
   - After distributing the 10 pairs, we have 7 red balls left to distribute into the 4 boxes. 
   - Each box must contain at least one red ball to ensure the number of red balls is greater than the number of white balls in each box.
   - We can use the stars and bars method again, but this time we need to ensure each box gets at least one red ball. We can do this by giving one red ball to each box first, and then distributing the remaining \( 7 - 4 = 3 \) red balls freely.
   - The formula for distributing \( n \) identical items into \( k \) distinct boxes with each box getting at least one item is given by:
     \[
     \binom{n-1}{k-1}
     \]
   - Here, \( n = 7 \) and \( k = 4 \). Therefore, the number of ways to distribute the remaining 3 red balls is:
     \[
     \binom{7-1}{4-1} = \binom{6}{3}
     \]

4. **Calculate the total number of ways**:
   - The total number of ways to distribute the balls is the product of the two combinations calculated above:
     \[
     \binom{13}{3} \cdot \binom{6}{3}
     \]
   - Calculate each binomial coefficient:
     \[
     \binom{13}{3} = \frac{13!}{3!(13-3)!} = \frac{13!}{3! \cdot 10!} = \frac{13 \cdot 12 \cdot 11}{3 \cdot 2 \cdot 1} = 286
     \]
     \[
     \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6!}{3! \cdot 3!} = \frac{6 \cdot 5 \cdot 4}{3 \cdot 2 \cdot 1} = 20
     \]
   - Multiply the results:
     \[
     286 \cdot 20 = 5720
     \]

The final answer is \(\boxed{5720}\).