We started off Math Circle today by talking about Scrabble, one of Shira's favorite games. Sometimes you get a great rack of letters, like:
Not a whole lot, right? (Actually for you Scrabble fans out there, there are a couple of two-letter words in this rack: AA, AE, AI and QI are all legal U.S. Scrabble words! Check out more here)
That's when we decided to invent our own language, a language that would make playing Scrabble super easy. In Houston math circle language, any combination of letters from the English alphabet, no matter what the length, no matter what order, counts as a word.
Yikes! We thought counting the number of SET sets we could make was hard- how on earth would we go about counting all the possible words?
It didn't take long for someone to say that this was impossible: we could make an infinite amount of words. After all, words could be of any length, so A, AA, AAA, AAAA, ..., infinite amount of As are all legal words! There's an infinite number of words right there, and we've only used one letter!
We decided to look at a more approachable question: how many 3-letter words can we make?
Well, we have 3 spots to fill: _____ _____ _____
and 26 choices for what to put in each spot (since there are 26 letters in the English alphabet)
We start by choosing the first letter: 26
For each one of those choices, we have 26 additional choices for what to choose as our second letter: 26 x 26
Then for each one of those choices, we have 26 more choices for our third letter: 26 x 26 x 26 = 263 = 17,576
Now that we knew what we were calculating, we went back to our secret language and started placing some restrictions. What if the 3-letter word had to start with A? What if it had to start and end with A? What if it just had to start and end with the same letter?