Factorials are pretty awesome. They introduce a cool new notation – I mean who knew that you would be using exclamation marks in maths! They are easy enough to understand at the basic level. They generate some mind-bogglingly large numbers really quickly, such is the power of repeated multiplication.
I like this as a way to start students off using their calculators with factorials.
The answer to the last one is, of course, 69! Be careful with that… It’s the largest number your calculator can calculate, why?
Factorials are used for various things, but probably the best place to start is combinations and permutations. I love this clip from QI and have shamelessly tried to be Stephen Fry in my classroom and tell my students how I am going to do something that no human has ever done before. I then shuffle the deck of cards…
Once you’ve finished the dramatic slamming of the cards on the table you can then start to discuss how we can find the number of possible ways those cards could be arranged, i.e. 52!
Then start calculating (using Standard Form) how many shuffles all the people who have ever lived on Earth, (around 100 billion) could have made if they made one every minute for their entire lives. Lots a crazy assumptions to get a very big number but still several orders of magnitude smaller than 52! Meaning that the chances are infinitesimally small that someone has shuffled that pack and got the exact same order before.
Or, just have fun with them as a thing. These questions can all be reasoned without a calculator and could be a good way to start before doing Permutations and Combinations.