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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