Theorem 3.4.7. Binomial Coefficient Formulas.

For \(n \in \mathbb{N}\text{,}\)
  1. \(\displaystyle \binom{n}{0} = 1\)
  2. \(\displaystyle \binom{n}{n} = 1\)
  3. \(\displaystyle \binom{n}{1} = n\)
  4. \(\displaystyle \binom{n}{n-1} = n\)
  5. \(\displaystyle \binom{n}{r} = \binom{n}{n-r}\)
  6. \(\displaystyle \binom{n+1}{r+1} = \binom{n}{r} + \binom{n}{r+1}\)

Proof.

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