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The Bell number, denoted is the total number of partitions of a set with elements. For , we have . For , we have
where are the Stirling numbers of the second kind.
Proof. We count the number of partitions of a set of  elements, depending on the size of the block containing the  st element. If the block has size  for
 then we have
 choices for the  other elements of the block. The remaining  elements can be partitioned in  ways. We have therefore that:

Using the formula above, one can easily derive the first few Bell numbers. Starting with , the first ten Bell numbers are 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147.
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