1. **Problem Statement:** We have two dice: the first with faces 1 to 6, the second with faces 1 to 10. Let $X$ be the sum of the numbers rolled on both dice. We want to find the value of the probability generating function (PGF) $G_X(2)$. 2. **Definition of Probability Generating Function:** The PGF of a discrete random variable $X$ taking values $k$ with probabilities $p_k$ is defined as: $$G_X(t) = E[t^X] = \sum_k p_k t^k$$ 3. **PGF of the sum of independent variables:** If $X = X_1 + X_2$ where $X_1$ and $X_2$ are independent, then: $$G_X(t) = G_{X_1}(t) \times G_{X_2}(t)$$ 4. **PGF of each die:** - For the first die (6 faces): $$G_{X_1}(t) = \frac{t^1 + t^2 + t^3 + t^4 + t^5 + t^6}{6}$$ - For the second die (10 faces): $$G_{X_2}(t) = \frac{t^1 + t^2 + t^3 + t^4 + t^5 + t^6 + t^7 + t^8 + t^9 + t^{10}}{10}$$ 5. **Calculate $G_X(2)$:** $$G_X(2) = G_{X_1}(2) \times G_{X_2}(2) = \left(\frac{2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6}{6}\right) \times \left(\frac{2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 + 2^9 + 2^{10}}{10}\right)$$ 6. **Evaluate powers of 2:** - $2^1=2$, $2^2=4$, $2^3=8$, $2^4=16$, $2^5=32$, $2^6=64$ - $2^7=128$, $2^8=256$, $2^9=512$, $2^{10}=1024$ 7. **Sum for first die:** $$2 + 4 + 8 + 16 + 32 + 64 = 126$$ 8. **Sum for second die:** $$2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 = 2046$$ 9. **Calculate each PGF value:** $$G_{X_1}(2) = \frac{126}{6} = 21$$ $$G_{X_2}(2) = \frac{2046}{10} = 204.6$$ 10. **Final value:** $$G_X(2) = 21 \times 204.6 = 4296.6$$ **Answer:** $$\boxed{4296.6}$$