Quick question about kids and numbers

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) \frac{11!}{4! 4! 2!}, 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?