2. Consider a list of n distinct numbers (a?,..., an). Recall that the MERGESORT algorithm sorts this list using O(n log n) comparisons. Can we do any better (i.e., can we sort a list using asymptotically fewer comparisons in the worst case)? In this problem, we will answer this question in the negative. (a) How many orderings are there of the n

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2. Consider a list of n distinct numbers (a?,..., an). Recall that the MERGESORT algorithm sorts this list using O(n log n) comparisons. Can we do any better (i.e., can we sort a list using asymptotically fewer comparisons in the worst case)? In this problem, we will answer this question in the negative. (a) How many orderings are there of the n numbers? (b) How many orderings are there of the n numbers where a? appears before a2? (c) How many orderings are there of the n numbers where a? appears before a2 and a3 appears before a4? (d) How many orderings are there of the n numbers where a? appears before a2 and a2 appears before az? (e) How many orderings are there of the n numbers where a? appears before a2 and a? appears before a3?

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