How many different Braille characters can there be?

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The Braille system for representing characters was developed in the early 19th century by Louis Braille. Special characters for the blind consist of raised dots. The positions for the dots are selected in two vertical columns of three dots each. There must be at least one raised dot. How many different Braille characters can there be?

In 1 you must look at the possibilities of the points being raised. Imagine you have a $2 \times 3$ grid that gives you the $6$ spaces. And you must divide your problem into $6$ disjoint sets, one represents the ways in which a point rises on the grid, set $2$ the ways in which two points rise on the grid and so on

I did this, but it's bad:\begin{align*} 3 \cdot 2 & = 6\\ 2^6 & = 64\\ \end{align*}$64 - 1 = 63$ characters.

I have this, but now I need to find the conjunte 2 3 4

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1 Answer

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There are $2^6$ subsets of $6$ positions, but one of these is empty, so the count is $2^6-1=63$. Alternatively, you can condition on the number $k$ of dots, yielding $$\sum_{k=1}^6 \binom{6}{k}=6+15+20+15+6+1=63$$

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