In Knuth's book "The art of computer programming" Vol 1, p.52, there is a picture with the caption "Geometric interpretation of $\binom{n+2}{3}$, $n=4$" (see attached photo).
I cannot figure out what it is. I did notice that there are $\binom{4+2}{3}=20$ dots and each dot is at the intersection of exactly 3 straight lines (and I don't know why some are dotted). Any suggestions?
On
I believe this is called the Pascal Pyramid of monomials. It has a base of n = 4, and the 3 refers to the dimension. See this image of Labeled Coefficents . It counts the number of terms that are possible of a degree $\leq$ n-1.
There are 20 combinations of 3 variables that are of a degree $\leq 4-1$.
On
There is a song called The Twelve Days of Christmas. If you go through it carefully, the total number of gifts accumulated at day number $n$ is your $$ \binom{n+2}{3} = \frac{(n+2)(n+1)n}{6}$$ In particular, at the end of the song, we have completed $n=12$ days, total gifts $$ \frac{14 \cdot 13 \cdot 12}{6} = 364 $$
Wikipedia says it probably began as a memory game for children, each child would try to recite part of it...
The exact origins and the meaning of the song are unknown, but it is highly probable that it originated from a children's memory and forfeit game.
The $n$th triangle number is the sum of the first $n$ positive integers. Therefore the $n$th triangle number is $$ \sum_{k=1}^n k = \frac{n(n+1)}{2} = \binom{n+1}{2}. $$ See this Wikipedia page for more information on triangle numbers.
The $n$th tetrahedral number is the sum of the first $n$ triangular numbers. The picture in Knuth's book is of the $4$th tetrahedral number: notice how each layer represents a triangle number. The $n$th tetrahedral number is $$ \sum_{k=1}^n \binom{k+1}{2} = \binom{n+2}{3}. $$ See this Wikipedia page for more information on tetrahedral numbers.