Hello zhupolongjoeYou sound a bit angry about this question, but I'll do my best!

First, let's make sure you understand what apartitionis. It's essentially a way of sharing out all the elements of a set intodisjointsubsets. By disjoint we mean 'having no elements in common'. So, for instance, in the second example that you are given, the set you're starting with has just two elements, the numbers 1 and 2. From this set {1, 2} you can form four possible subsets, which are:

(the empty set), {1}, {2} and {1, 2} (the set itself)

Now when wepartitionthe set, we take one or more of these subsets, according to the following rules:

- the empty set is not allowed
- no two subsets may have any elements in common - they're disjoint, remember
- all the elements of the original set must be included somewhere

So there are two ways of doing this. They are:

- {1}, {2}; and
- {1, 2}

So, in the notation used in the question, = 2.

OK, so now let's look at the set {1, 2, 3} and work out what is; that's the number of ways in which we can partition this set.

First we could have subsets with just one element each:

- {1}, {2}, {3}

Then we could have one subset with one element and one subset with two elements. There are three ways of doing this:

- {1}, {2, 3}
- {2}, {1, 3}
- {3}, {1, 2}

Finally, we could have all three elements in the subset which is simply the set itself:

- {1, 2, 3}

So there are 5 ways of partitioning this set altogether. So .

So, what have you got to do now? First, repeat the sort of thing I've just done here with the set {1, 2, 3, 4}. Do it in a logical way, similar to what I've done here, starting with subsets with just one element each, and working up to the subset containing all four elements. You should find 15 ways of partitioning this set. So = 15.

Then, look for a way of describing how this will work in the general case with the set {1, 2, ..., n}. This should give you the answer you're looking for.

Best of luck!

Grandad