Is Each Stage of Pascal's Triangle Special?

[Agent Smith]

Vide supra is a first few stages of Pascal's Triangle. The numbers themselves are the coefficients for the binomial expansion of $(a + b)^n$, where $n \in \{0, 1, 2, ... \}$. I'm just curious whether each stage has a special meaning, unique to it. My lack of knowledge prevents me from giving examples; my best shot is that $(a + b)^0 = 1$, we know $1$ is special, it's the foundation of counting and hence arithmetic. I hope that suffices to get my point across.

I've highlighted the 6th stage of Pascal's triangle with a red rectangle and consists of the coefficients of the terms of $(a + b)^6 = 1a^6b^0 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6a^1b^5 + 1a^0b^6$ because I'd like to know if $(a + b)^6$ is special in any way e.g. (pure speculation) the coefficients give us the number of dimensions in which the Pythagorean theorem is false.

Gracias.

 

[FactChecker]

The individual entries on a row are the binomial coefficients. They occur in many situations. One is in calculating the number of combinations of two possible alternatives. That is important in probability and statistics. See this: https://en.wikipedia.org/wiki/Binomial_coefficient#Combinatorics_and_statistics

 

[Svein]

Look at the triangle. Observe that each element is the sum of the two elements above it (in your row, 6=1+5, 15=5+10, 20=10+10 etc.).

 

[fresh_42]

The elements in the triangle are all of them form $\binom n k$ which represents the number of possible selections of $k$ elements from a set with $n$ elements.

That means you have all possible interpretations of such selections plus all possible formulas of binomial coefficients at hand. I think that's it.

 

[Mark44]

I'd like to know if $(a + b)^6$ is special in any way e.g. (pure speculation) the coefficients give us the number of dimensions in which the Pythagorean theorem is false.

The rows themselves aren't special in any way.

 

[FactChecker]

I'd like to know if $(a + b)^6$ is special in any way e.g. (pure speculation) the coefficients give us the number of dimensions in which the Pythagorean theorem is false.

The Pythagorean Theorem is about the lengths of the sides of a triangle. In any n-dimensional Euclidean space, the triangle is always on a two dimensional plane, so the Pythagorean Theorem reduces to the two dimensional case regardless of the value of n. The curvature of a space in which a "triangle" (three "straight" lines) is embedded is more important to the validity of the Pythagorean Theorem.

 

[Svein]

Also observe that $2^{n} = (1+1)^{n}=\sum_{i=0}^{n}\begin{pmatrix} n \\ i\\ \end{pmatrix} 1^{n-i}\cdot 1^{i}=\sum_{i=0}^{n}\begin{pmatrix} n \\ i\\ \end{pmatrix}$

 

[Agent Smith]

The individual entries on a row are the binomial coefficients. They occur in many situations. One is in calculating the number of combinations of two possible alternatives. That is important in probability and statistics. See this: https://en.wikipedia.org/wiki/Binomial_coefficient#Combinatorics_and_statistics

Apologies for the late reply, got caught up with work.

First off gracias.

Second, I should've been more direct. Is there anything special about $(a + b)^6$, that makes it unique from other $(a + b)^n$ where $n \ne 6$? Mathematics has branched out into so many subdisciplines like real analysis, topology, knot theory, and so on. Are the numbers (the coefficients) $1, 6, 15$ in any way important in one/more of these fields?

From the little that I know ...
$1$ is unity, our basic counting entity.
$6$ is the first perfect number (?)
$15$ is the smallest semiprime with both factors being odd
$20$ is ... ???

We can see that each number can be (is) unique but I would like a common thread that unites them, in that $\{1, 6, 15, 20\}$ is the solution set to a problem, other than find the coefficients of the binomial expansion of $(a + b)^6$.

Look at the triangle. Observe that each element is the sum of the two elements above it (in your row, 6=1+5, 15=5+10, 20=10+10 etc.).

That I'm familiar with and now that you've reminded me of it, I also recall that each element number at a point in Pascal's triangle is the number of unique paths by which that point may be arrived at. So 6 is the point that has 6 unique pathways to it. Would you like to comment on that score?

The elements in the triangle are all of them form $\binom n k$ which represents the number of possible selections of $k$ elements from a set with $n$ elements.

That means you have all possible interpretations of such selections plus all possible formulas of binomial coefficients at hand. I think that's it.

Si, verum. However, I was told, in math a particular numerical fact/theorem/system has multiple applications. A geometric theorem could help solve a problem in number theory (throwing darts here!). Please read my replies to other comments (vide supra).

The rows themselves aren't special in any way.

I wonder why.

The Pythagorean Theorem is about the lengths of the sides of a triangle. In any n-dimensional Euclidean space, the triangle is always on a two dimensional plane, so the Pythagorean Theorem reduces to the two dimensional case regardless of the value of n. The curvature of a space in which a "triangle" (three "straight" lines) is embedded is more important to the validity of the Pythagorean Theorem.

That was just lorem ipsum. A, in this case nonsensical, filler until the real McCoy arrives. Please read my replies above.

 

[Agent Smith]

Also observe that $2^{n} = (1+1)^{n}=\sum_{i=0}^{n}\begin{pmatrix} n \\ i\\ \end{pmatrix} 1^{n-i}\cdot 1^{i}=\sum_{i=0}^{n}\begin{pmatrix} n \\ i\\ \end{pmatrix}$

Aah $(a + b)^n$ when $a = b = 1$?

 

[jbriggs444]

So 6 is the point that has 6 unique pathways to it.

Obviously. If the number of paths to the above-and-left parent is $m$ and the number of paths to the above-and-right parent is $n$ then the number of paths to the node in question is $m+n$.

This line of inquiry feels like a solution in search of a problem.

 

[Agent Smith]

This line of inquiry feels like a solution in search of a problem.

Interesting you say that.
Gracias

 

[fresh_42]

However, I was told, in math a particular numerical fact/theorem/system has multiple applications. A geometric theorem could help solve a problem in number theory (throwing darts here!).

See post #7. The cardinality of the power set of a set with $n$ elements is $2^n.$

Please read my replies to other comments (vide supra).

To find what? Your comments are far too vague. Google Pascal's triangle and you will find hundreds of pages. The answer to your question is combinatorics. Everything else is related to combinatorics like post #7 or fishing in the dark.

The rows themselves aren't special in any way.

This line of inquiry feels like a solution in search of a problem.

The question has been answered and this thread is closed now.