[Math] How many ways to divide a n-element set

Question

For a n-element set, how many ways to divide the set to a combination of subsets?

For example, for the set {1, 2, 3}, there would be 5 ways to divide:

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

For the n-element set, I thought it would be

(n^n)/n!

, because imagine dividing as drop n balls to n indistinguishable boxes. But apparently, it is incorrect in the case n=3 .

Answer 1

You want the Bell numbers --- link to Wikipedia entry.