FactChecker said:
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##.
Svein said:
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?
fresh_42 said:
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).
Mark44 said:
The rows themselves aren't special in any way.
I wonder why.
FactChecker said:
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.