1. **Stating the problem:** Find the sum of the first thirteen terms of the exponential sequence with first term $a_1 = \frac{10 \cdot \binom{11}{6}}{7}$ and common ratio $r = \frac{1}{2}$. 2. **Recall the formula for the sum of the first $n$ terms of a geometric sequence:** $$S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$$ where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. 3. **Calculate the first term $a_1$:** $$a_1 = \frac{10 \cdot \binom{11}{6}}{7} = \frac{10 \cdot 462}{7} = \frac{4620}{7} = 660$$ 4. **Substitute values into the sum formula:** $$S_{13} = 660 \cdot \frac{1 - \left(\frac{1}{2}\right)^{13}}{1 - \frac{1}{2}}$$ 5. **Simplify the denominator:** $$1 - \frac{1}{2} = \frac{1}{2}$$ 6. **Calculate the numerator:** $$1 - \left(\frac{1}{2}\right)^{13} = 1 - \frac{1}{8192} = \frac{8191}{8192}$$ 7. **Calculate the sum:** $$S_{13} = 660 \cdot \frac{\frac{8191}{8192}}{\frac{1}{2}} = 660 \cdot \frac{8191}{8192} \cdot 2 = 1320 \cdot \frac{8191}{8192}$$ 8. **Approximate the sum:** $$S_{13} \approx 1320 \times 0.99987793 = 1319.84$$ **Final answer:** $$\boxed{S_{13} = 1320 \cdot \frac{8191}{8192} \approx 1319.84}$$