1. **State the problem:** Simplify the expression $$\frac{p}{2} - (q + p - (2 + q))$$ and then substitute $p=2$ and $q=10$. 2. **Write the expression:** $$\frac{p}{2} - (q + p - (2 + q))$$ 3. **Simplify inside the parentheses:** Inside the parentheses, simplify $q + p - (2 + q)$. 4. **Distribute the minus sign:** $$q + p - (2 + q) = q + p - 2 - q$$ 5. **Cancel terms:** $$q - \cancel{q} + p - 2 = p - 2$$ 6. **Rewrite the original expression:** $$\frac{p}{2} - (p - 2)$$ 7. **Distribute the minus sign:** $$\frac{p}{2} - p + 2$$ 8. **Combine like terms:** Rewrite $p$ as $\frac{2p}{2}$ to combine with $\frac{p}{2}$: $$\frac{p}{2} - \frac{2p}{2} + 2 = \frac{p - 2p}{2} + 2 = \frac{-p}{2} + 2$$ 9. **Substitute $p=2$:** $$\frac{-2}{2} + 2 = -1 + 2 = 1$$ **Final answer:** $$1$$