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Given this description of the elements in Pascal's triangle, we can rewrite the above observations as follows:

1. \({n \choose 0} = 1\) and \({n \choose n} = 1\text{.}\)

2. \({n \choose k} = {n-1 \choose k-1} + {n-1 \choose k}\text{.}\)

3. \({n \choose k} = {n \choose n-k}\text{.}\)

4. \({n\choose 0} + {n \choose 1} + {n \choose 2} + \cdots + {n \choose n} = 2^n\text{.}\)