Question

The number of ways in which n different things can be arranged into r different groups

Answer 1

Answer:

The number of ways to partition a set of n things into r nonempty subsets is known as a Stirling number of the second kind.  There are various notations; most modern writers seem to be pushing for the notation

$\displaystyle\left\{\begin{array}{c}n\\r\end{array}\right\}$

These are similar to binomial coefficients, as the notation suggests.

Just as you can calculate binomial coefficients using

$\displaystyle\binom{n}{r}=\binom{n-1}{r}+\binom{n-1}{r-1},\quad\binom{n}0=\binom{n}{n}=1$

you can calculate Stirling numbers of the second kind using

$\displaystyle\left\{\begin{array}{c}n\\r\end{array}\right\}=r\left\{\begin{array}{c}n-1\\r\end{array}\right\}+\left\{\begin{array}{c}n-1\\r-1\end{array}\right\},\quad\left\{\begin{array}{c}n\\0\end{array}\right\}=0,\quad\left\{\begin{array}{c}n\\n\end{array}\right\}=1$