Binomial Choices (Artwork)
Binomial Choices Digital Print 2021

All patterns of choosing items from a set of ten, from zero ( far left) to ten (far right). The patterns are grouped into the 11 possibilities for the number of choices in for a given size. The number in each group is a binomial coefficient related the number of ways to choose \(k\) items from a set of 10 items, resulting in a binomial distribution.

Background and Inspiration

This piece illustrates the way binomial coefficients relate to counting the number of ways of choosing \(k\) items from a set of \(n\) items.

The number of choosing \(k\) items from a set of \(n\) items is given by row \(n\) of Pascal's triangle (see below). This artwork corresponds to \(n = 10\). When \(k\) is 0 or 10, there is only 1 possible pattern (no colors). When \(k\) is 1 or 9, there are 10 possible patterns (all colors). For \(k\) or \(10-k\), equaling 2, 3, 4, and 5, the number of possible patterns are 45, 120, 210, and 252 respectively.

Pascal's triangle with 16 rows.

Each row in the column with \(k\) colors can be thought of a set, with \(10 \choose k\) subsets of the set with \(10\) colors. Thus, there are \(2^{10} = 1024\) total subsets that can be made from a set with \(10\) colors.

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