# Explanation: Properties of the Pascal's Triangle

There are lots of beautiful properties of the Pascal's Triangle %(blue)(feel free to add a new property if you know about another by clicking the edit button)%:

$$(1)$$ Each row of the Pascal's Triangle begins with $$1$$'s.

$\begin{array}{c|ccccccccccccc} n&\binom n0&\binom n1&\binom n2&\binom n3&\binom n4&\binom n5&\binom n6&\binom n7&\binom n8&\binom n9&\binom n{10}&\binom n{11}&\cdots\\ \hline 0&\fbox 1&&&&&&&&&&\\ 1&\fbox 1&1&&&&&&&&&\\ 2&\fbox 1&2&1&&&&&&&&\\ 3&\fbox 1&3&3&1&&&&&&&\\ 4&\fbox 1&4&6&4&1&&&&&&\\ 5&\fbox 1&5&10&10&5&1&&&&&\\ 6&\fbox 1&6&15&20&15&6&1&&&&\\ 7&\fbox 1&7&21&35&35&21&7&1&&&\\ 8&\fbox 1&8&28&56&70&56&28&8&1&&\\ 9&\fbox 1&9&36&84&126&126&84&36&9&1&\\ 10&\fbox 1&10&45&120&210&252&210&120&45&10&1\\ 11&\fbox 1&11&55&165&330&462&462&330&165&55&11&1\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots \end{array}$

$$(2)$$ The second numbers in each row form the natural numbers. $\begin{array}{c|ccccccccccccc} n&\binom n0&\binom n1&\binom n2&\binom n3&\binom n4&\binom n5&\binom n6&\binom n7&\binom n8&\binom n9&\binom n{10}&\binom n{11}&\cdots\\ \hline 0&1&\fbox 0&&&&&&&&&\\ 1&1&\fbox 1&&&&&&&&&\\ 2&1&\fbox 2&1&&&&&&&&\\ 3&1&\fbox 3&3&1&&&&&&&\\ 4&1&\fbox 4&6&4&1&&&&&&\\ 5&1&\fbox 5&10&10&5&1&&&&&\\ 6&1&\fbox 6&15&20&15&6&1&&&&\\ 7&1&\fbox 7&21&35&35&21&7&1&&&\\ 8&1&\fbox 8&28&56&70&56&28&8&1&&\\ 9&1&\fbox 9&36&84&126&126&84&36&9&1&\\ 10&1&\fbox {10}&45&120&210&252&210&120&45&10&1\\ 11&1&\fbox {11}&55&165&330&462&462&330&165&55&11&1\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots \end{array}$

$$(3)$$ The third numbers in each row form the triangle numbers. $\begin{array}{c|ccccccccccccc} n&\binom n0&\binom n1&\binom n2&\binom n3&\binom n4&\binom n5&\binom n6&\binom n7&\binom n8&\binom n9&\binom n{10}&\binom n{11}&\cdots\\ \hline 0&1&0&&&&&&&&&\\ 1&1&1&&&&&&&&&\\ 2&1&2&\fbox 1&&&&&&&&\\ 3&1&3&\fbox 3&1&&&&&&&\\ 4&1&4&\fbox 6&4&1&&&&&&\\ 5&1&5&\fbox {10}&10&5&1&&&&&\\ 6&1&6&\fbox {15}&20&15&6&1&&&&\\ 7&1&7&\fbox {21}&35&35&21&7&1&&&\\ 8&1&8&\fbox {28}&56&70&56&28&8&1&&\\ 9&1&9&\fbox {36}&84&126&126&84&36&9&1&\\ 10&1&10&\fbox {45}&120&210&252&210&120&45&10&1\\ 11&1&11&\fbox {55}&165&330&462&462&330&165&55&11&1\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots \end{array}$

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