In communicating your solutions, please describe any conditions or assumptions you made as you explored and solved the problems.

#1

  In the Braille system, a symbol (such as a lowercase letter, punctuation mark, suffix, and so on) is given by raising at least one of the dots in the six-dot arrangement shown in figure (a). [The six Braille positions are labeled in figure (a).] For example, in (b), the dots in positions 1, 3, and 4 are raised and this six-dot arrangement represents the letter m. The definite article the is shown in (c), and the semicolon (;) is given by the six-dot arrangement in (d), where the dots in positions 2 and 3 are raised.


a) How many different symbols can be represented in the Braille system?

b) How many symbols have an odd number of raised dots?

c) How many symbols have no more than one raised dot in each column, where dots 1, 2, and 3 are in the same column, and dots 4, 5, and 6 are in the same column?

#2

Jackie is a Scrabble® player. She wanted to know the total number of potential two- and three-letter words she could make. Here is her reasoning:


There are  26€26=676  possible two-letter words and   26€26€26=17,576 possible three-letter words.  Therefore, there are (26€26)(26€26€26)=11,881,376 possible words in all.

           Is Jackie right? Explain.