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was studying the book "Principles and Techniques in Combinatorics" and I came across the following question from Chapter 1 that I can not solve. I would if possible give an explanation and or a solution to such an issue.

Question: (Chapter 1. Permutations and Combinations, p. 23, exercise 1, question 27) Let $S = \{1, 2, . . . , n + 1\}$ where $n\geq 2$, and let $T=\{(x,y,Z)\in S^3; x< z \mbox{ and } y<z\}$. Show by counting $|T|$ in two different ways that $\displaystyle\sum_{k=1}^n k^2=|T|= \begin{pmatrix}n+1\\ 2 \end{pmatrix}+ 2\begin{pmatrix}n+1\\ 3 \end{pmatrix}$.

Thank you very much.

Kind regards.

Emanuel Viana
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    Almost a duplicate of https://math.stackexchange.com/questions/3113187/combinatorial-proof-of-sum-k-1n-k2-binomn13-binomn23/ – Mike Earnest Apr 10 '19 at 00:04

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The proof technique the book is trying to lead you to use is called Proof by double counting. The idea being, if you have a scenario you wish to count and via different methods you arrive at two different expressions which correctly count the scenario, then the expressions must in fact be equal.

Hints:

Left side:

Break into cases based on the value of $z$. If $z=k$, how many choices are available for $x$? How many choices are available for $y$?

Right side:

Break into cases based on which of $x<y<z$ or $y<x<z$ or $x=y<z$ is true. Pick the two or three values seen simultaneously.

JMoravitz
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