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  1. \(\displaystyle \binom{n}{0} = \frac{n!}{0!(n-0)!} = 1\)
  2. \(\displaystyle \binom{n}{n} = \frac{n!}{n!(n-n)!} = 1\)
  3. \(\displaystyle \binom{n}{1} = \frac{n!}{1!(n-1)!} = n\)
  4. \(\displaystyle \binom{n}{n-1} = \frac{n!}{(n-1)!(n-(n-1))!} = n\)
  5. \(\displaystyle \binom{n}{r} = \frac{n!}{r!(n-r)!} = \frac{n!}{(n-r)!(n-(n-r))!} = \binom{n}{n-r}\)
  6. \begin{align*} \binom{n}{r} + \binom{n}{r+1} & = \frac{n!}{r!(n-r)!} + \frac{n!}{(r+1)!(n-(r+1))!}\\ & = (r+1) \frac{n!}{(r+1)!(n-r)!} \\ + (n-r) \frac{n!}{(r+1)!(n-r))!}\\ & = \frac{(r+1) n! + (n-r)n!}{(r+1)!(n-r)!}\\ & = \frac{(n+1) n!}{(r+1)!((n+1)-(r+1))!}\\ & = \binom{n+1}{r+1} \end{align*}
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