The forum discussion centers on proving the combinatorial identity $$\binom{n}{2k+1}=\sum_{i=1}^n{\binom{i-1}{k}\binom{n-i}{k}}$$ using the median method. Participants explore the classification of (2k + 1)-sets of size n by their median, emphasizing the division of the set into elements smaller and larger than the median. The discussion highlights the relationship between choosing elements from subsets and the overall combinatorial structure, ultimately leading to the conclusion that the left-hand side represents choosing 2k elements from n-1 elements.
PREREQUISITESMathematics students, combinatorial theorists, and educators seeking to deepen their understanding of combinatorial proofs and identities.
But I don't know how to do that, or what to do from there.Classify (2k + 1)-sets of size n by what the median is, where (2k + 1)-sets of size n means the subsets of size n containing (2k + 1) elements.
Office_Shredder said:I think the hint is a bit poorly worded, but here's the idea:
On the left, in a set of size n we are picking 2k+1 elements. Consider the set to literally be the numbers 1 through n. Split your set into two parts - the numbers smaller than the median of the 2k+1 numbers you picked, and the numbers larger than it.
On the other hand, each term in the summand is counting the ways to: pick k elements from an i-1 sized set, and pick k elements from an n-i sized set. If we combine those two together, we are essentially picking 2k elements from a set of size n-1.