## Sums of binomial coefficients

### Task number: 3418

Prove algebraically and also using combinatorial reasoning:

#### Variant

\(\binom{n-1}{k-1}+\binom{n-1}{k}=\binom{n}{k}\)

#### Variant

\( \sum\limits_{k=0}^n {n \choose k} = 2^n\)

#### Variant

\( \sum\limits_{k=0}^n (-1)^k \binom{n}{k} = 0\)

#### Variant

\( \binom{n}{m}\binom{m}{r}=\binom{n}{r}\binom{n-r}{m-r}\)

#### Variant

\( \sum\limits_{k=0}^r \binom{n}{k}\binom{m}{r-k} = \binom{m+n}{r}\)

#### Variant

\( \sum\limits_{k=r}^n \binom{k}{r} = \binom{n+1}{r+1} \)