This morning in our little summer tour through Art of Problem Solving’s *Introduction to Counting and Probability* we came across the question:

How many different ways are there to arrange the letters of the word MISSISSIPPI?

The question has a couple of potential ways to leave the answer. You could write:

(1) , or

(2) 34,650

for example. However, the boys told me that the answer was:

(3) 11*10*9*7*5.

So, they were sort of in the middle of (1) and (2) – not wanting to leave the factorials, but also not wanting to do the multiplication. Ha!

I actually wanted them to work out the multiplication simply to continue to build up their numbers number sense. At the same time, though, answer (1) above is really want you want to build up their counting sense (for lack of a better term).

So I was just curious what people thought was the best way to express the answer for this question? Also, I suppose, if the answer to my first question would vary based on the age of the students working on the problem?

Thanks!

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My kids are a bit younger than yours, but if they were doing something and had an answer 11*10*9*7*5, I would ask if they could approximate it, expecting them to eventually get to it being close to 35,000. Then, we could do some experiments to see if it would be more or less (basically comparing (a+x)*(a-x) vs a^2).

I know your problem just wanted to end with the number of arrangements, but if there was a further application, the form your sons chose could be really helpful, since it is so close to a prime factorization. Explicitly writing the prime factorization would be another form of answer and really easy extension for them.

One of the happiest decisions I made teaching the kids math was to go through AoPS’s Intro to Number Theory with both of them. That work really helped build up their number sense.

One other idea is to use this opportunity to encourage their sense of mathematical aesthetics. What format do they prefer and why? If they are engaged in the conversation, I bet you will hear some really interesting things about how they understand and feel about certain patterns.