1. **Problem Statement:** Compare the time complexities of BottomUpSort (Merge Sort) and SelectionSort for $n = 32,768$. 2. **Formulas and Rules:** - BottomUpSort (Merge Sort) has time complexity $O(n \log n)$. - SelectionSort has time complexity $O(n^2)$. - We will calculate approximate values for $n \log n$ and $n^2$ to compare. 3. **Calculate $n \log n$:** - Here, $n = 32,768$. - Since $32,768 = 2^{15}$, $\log_2 32,768 = 15$. - So, $n \log n = 32,768 \times 15 = 491,520$. 4. **Calculate $n^2$:** - $n^2 = (32,768)^2 = 1,073,741,824$. 5. **Comparison:** - BottomUpSort time complexity is approximately $491,520$ operations. - SelectionSort time complexity is approximately $1,073,741,824$ operations. - SelectionSort takes significantly more time than BottomUpSort for $n=32,768$. 6. **Conclusion:** - BottomUpSort (Merge Sort) is much more efficient than SelectionSort for large $n$ due to its $O(n \log n)$ complexity compared to $O(n^2)$ of SelectionSort.