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| rev | Admin | (Created page with "Prove that if <math>n \ge 0</math>, then <math display="block"> \sum_{k = 0}^n {n \choose k}^2 = {{2n} \choose n}\ . </math> '' Hint'': Write the sum as <math display="block"> \sum_{k = 0}^n {n \choose k}{n \choose {n-k}} </math> and explain why this is a coefficient in the product <math display="block"> (1 + x)^n (1 + x)^n\ . </math> Use this, together with Exercise, to show that <math display="block"> u^{(2)}_{2n} = \frac 1{4^{2n}}{{2n}\ch...") | Jun 15'24 at 1:45 | +561 |