TITLE: Bell numbers (number of partitions of set of cardinality n) recurrence relation proof
QUESTION [7 upvotes]: Let $X$ be a set of cardinality $n$. How many partitions does it have? The users on the website found that these are the so called bell numbers. They also pointed out the following recurrence relation:
$$B_{n+1}=\sum\limits_{k=0}^n\binom{n}{k}B_k$$
Could someone provide some insight and prove this?

REPLY [2 votes]: This is essentially the answer of Austin Mohr, but skipping the step of symmetry. I personally find it easier to remember this way of arguing.

To partition the set $\{0,\dots,n\}$, we proceed as follows:

Decide how many elements $k$ we wish not to be contained in the part of $0$; there are $\binom nk$ options;
Choose a partition of those $k$ elements; there are $B_k$ options.

Summing over all $k$ gives $B_{n+1}$. Thus $$B_{n+1}=\sum_k\binom nk B_k.$$