Prove-that-n-1-C-r-n-C-r-n-C-r-1-

Question Number 197311 by cortano12 last updated on 13/Sep/23

Prove that

_{n + 1} C_{r} =_{n} C_{r} +_{n} C_{r - 1}

Answered by qaz last updated on 14/Sep/23

(\begin{matrix} n + 1 \\ r \end{matrix}) = (\begin{matrix} n \\ r \end{matrix}) + (\begin{matrix} n \\ r - 1 \end{matrix}) ?

(\begin{matrix} n + 1 \\ r \end{matrix}) = [x^{r}] {(1 + x)}^{n + 1}

= [x^{r}] ({(1 + x)}^{n} + x {(1 + x)}^{n})

= (\begin{matrix} n \\ r \end{matrix}) + (\begin{matrix} n \\ r - 1 \end{matrix})

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