A combinatorial proof of the following relation: $$\binom{n+2}{3} = 1 \cdot n + 2 \cdot (n-1) + 3 \cdot (n-2) + \cdots + n \cdot 1$$
So one side is the number of ways to choose $3$ numbers from $n+2$ numbers and the other side I did like this:
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4Your combinatorial proof seems fine. – Clement Yung Feb 16 '22 at 11:49
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2Clever! I'd state it as something like: first choose the middle-largest of the three numbers, then choose the other two. – Karl Feb 16 '22 at 12:16