Number of ways to divide N different points into k groups

Question

I'm trying to calculate the following. Given N different points in 2D space, and $1 \leq K \leq N $

a) The number of possibilities, to divide the points into K groups, without importance of the group numbers.

b)The number of possibilities, to divide the points into K groups, with importance of the group numbers.

Empty groups are not allowed.

Thx for the help :)

Cheers

Answer 1

You are looking for:

a) Stirling numbers of the second kind
b) a)$\times k!.$ This are related to Fubini numbers. Notice that the $k!$ what it does is permutes the "group numbers".