Even though it seems a little far-fetched I will use the Binomial Theorem. The definition of number $\binom{n}{r}$ has a reason to come to exist in the development of $(x + y)^n$. So I think the natural and instructive. For all $x,y\in\mathbb{R}$ we have,
\begin{array}{rrl}
\hspace{2cm}&( x+y)^{n+1}= & (x+y)\cdot ( x+y)^n,
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\Longleftrightarrow & \sum_{k=0}^{n+1}\binom{n+1}{k}x^{(n+1)-k}y^{k}= & (x+y)\cdot \sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^{k},
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\Longleftrightarrow & \sum_{k=0}^{n+1}\binom{n+1}{k}x^{(n+1)-k}y^{k}= & \sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^{k+1}+ \sum_{k=0}^{n}\binom{n}{k}x^{n-k+1}y^{k}.
\end{array}
Then for all $k=1,2,\ldots n$ we have
\begin{array}{rl}
\binom{n+1}{k}x^{(n+1)-k}y^{k}= & \binom{n}{(k-1)}x^{n-(k-1)}y^{(k-1)+1}+ \binom{n}{k}x^{n-k+1}y^{k}.
\end{array}
For $x=1$ and $y=1$,
\begin{array}{rl}
\binom{n+1}{k}= & \binom{n}{(k-1)}+ \binom{n}{k},\qquad k=1,2,\ldots n.
\end{array}
Setting $k = r +1$ then
\begin{array}{rl}
\binom{n+1}{r+1}= & \binom{n}{r}+ \binom{n}{r+1},\qquad r=0,1,2,\ldots n-1.
\end{array}