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Binomial coefficients

  1. Evaluate the following sums using combinatorial methods and algebraic methods :

    • \(\displaystyle \sum_{i=0}^{k} {m \choose i}{n \choose k-i}\)

    • \(\displaystyle \sum_{i=0}^{n} {n \choose i}^2\)

    • \(\displaystyle \sum_{i=0}^{n} (-1)^i{n \choose i}\)

    • \(\displaystyle \sum_{k=0}^{n}k {n\choose k}\)

  2. Let \(n>1\) be an odd integer. Prove that the following sequence contains an odd number of odd numbers.

\({n \choose 1}, {n \choose 2}, \dots, {n \choose \frac{n-1}{2}}\)

Source: folklore