00:01
For this problem, we've got some counting problems.
00:04
Number one, we've got eight people sitting in a round table.
00:10
So let's just draw ourselves a table here, and we've got a bunch of numbered chairs.
00:15
One, two, three, four, five, six, seven, eight.
00:21
And we've got eight people.
00:23
So let's start with part a, where everything is numbered.
00:26
So we want to figure out how many people we can, how many ways we can sit these eight people in these eight numbered chairs.
00:33
Now, since the chairs are numbered, that means that, for example, if you had someone sitting in chair one and the person sitting in chair two and so on all the way around, that would be different from person one sitting in chair two and then the person next to them sitting in chair three and so on.
00:51
So basically, the chair you're sitting in matters for this problem.
00:56
So if we look at chair one, how many choices are there? well, we've got eight people, so we can seat eight different people in chair number one.
01:05
So that gives us eight choices for chair number one.
01:08
And then for chair number two, there's seven remaining people.
01:11
So there's seven choices for chair two and six choices for chair three and so on.
01:19
So that means we've got eight times seven times six all the way down to one choices, which is otherwise known as eight factorial.
01:31
And this is the total number of ways you can arrange people in these numbered chairs.
01:36
So let's just see what eight factorial is.
01:39
That's quite a large number, 40 ,320.
01:44
Okay, and now part b.
01:46
This time, we imagine that our chairs aren't numbered.
01:51
So that would mean that a configuration of people is the same if you're just rotating the table, basically.
02:10
So if you have person one in chair one and person two in chair two and so on, that's the same as person one being in chair two, person two being in chair three, and person three being in chair four and so on.
02:20
So this time, there's a little bit more.
02:23
Some of the options are counted twice, basically.
02:27
So how can we solve this problem? well, we can think about it this way.
02:35
Imagine you've got person number one sitting in any chair.
02:39
It doesn't matter.
02:40
Then how many choices are there for the person who sits to the right of them? well, there are seven people left, so there are seven choices.
02:49
So seven people could possibly sit to the right of person number one.
02:52
And now that you have person...
02:54
Now that you've chosen the person who's sitting to the right of person one, you can do the same for person number two for those six choices now.
03:04
So six choices for person number two, and all the way down to one, just like part a, except this time we start at seven because the numbers on the chairs aren't there anymore.
03:17
So this time, it's just seven factorial...