Dominated Turtle

Two turtles, Tort and Bort, are going to perform independent simple symmetric random walks on the integers starting at positions $0$ and $4$, respectively. Compute the probability after $10$ steps, Tort and Bort are back at their initial positions and that Tort was strictly behind Bort at all $10$ steps. The answer will be in the form

$$\dfrac{\displaystyle \binom{a}{b}^p - \binom{c}{d}^p}{2^r}$$

where $b \leq \dfrac{a}{2}$ and $d \leq \dfrac{c}{2}$. Find $a^2 + b^2 + c^2 + d^2 + p^2 + r^2$.