## ATIKE and EZO

Earlier this week I noticed that Katie was scrambling the letters in her name for fun.  This seemed to me like a perfect excuse to do some combinatorics with her.

Me: Lets see how many different “words” we can make out of the letters in Zoe’s name.

Katie: Why Zoe?

Me: Because there are only three letters.

This seemed convincing enough to her.  After writing down four possibilities, she paused.  I pointed out to her that she had two starting with the letter a and only one for each of o and e.  This was enough for her to quickly figure out the missing two.  I then went on to explain to her how we know that we have all of them by saying that for each starting letter there are two possible ways of writing down the remaining two letters.

Katie: Ok, let’s do Ari now.

Me: How many do you think we’ll get?

Katie: Six, because there are also three letters.

Me: What if we were to do катя (Katie in Russian), would we get more or fewer?

Katie: More, maybe 9…

Me: We’d actually get 24.  What about веня (Ben in Russian)?

Katie:  Also 24.  Hey, we get 2 pairs: zoe and ari, катя and веня.

She then went through a long list of relatives’ names, placing them all into groups (luckily, almost all of them had three or four letters in them).  Some four-letter names had a repeating letter and I told her that those will have a different number of permutations, so taking that on faith she put them into a separate group.

This morning I noticed Katie writing down all the permutations of ari, completely on her own initiative, so this evening I suggested to her that we write down all of the permutations of катя.  Here is the result:

My only advice was to group together all of the “words” that started with the same letter.  The rest she did on her own.  It was also interesting to see the technique that she used for writing the words as she progressed through the different first letters.  For the first two it looked like she wrote them down randomly, but for the last two, I could tell that she was doing them systematically.  Needless to say that she had a lot of fun reading all of them once she was done.