Distributing $19$ different presents to $6$ children so that each has at least $2$ presents? In how many ways can you distribute $19$ different presents among $6$ children so each can get at least $2$ presents?
 A: You need the $2-$ associated Stirling numbers of the second kind. The idea is that to give out the presents you can first separate the presents into six groups of size at least two and then chose which group you want to give to each children (Clearly once the presents have been seperated into groups there are $6!$ ways to do this.
But how many ways are there to seperate the $17$ presents into $6$ groups of size at least $2$? This number is precisely the $2-$ associated Stirling number of the second kind $S_2(17,6)$. The $2-$ associated Stirling numbers of the second kind can be found in this OEIS sequence http://oeis.org/A008299.
You can download the text file with a lot of values here although finding the exact term may be slightly tricky. you should be able to find $S_2(19,6)$ which is $254752658160$. So what you want is $6!\times 254752658160= 183421913875200$
Also note that you can calculate the values of $S_2(n,k)$ yourself  by using the recurrence $S_2(n+1,k)=S_2(n,k)+nS_2(n,k-1)$
The explanation for the recurrence is simple. Pick a fixed present $a$. There aare two types of distributions possible, those in which the group of $a$ still has at least two presents after removing $a$ (there are $S_2(n,k)$ of these since removing $a$ gives a valid partition of $n$ presents in $k$ groups of at least two presents each). The other type of partition is that in which removing present $a$ leaves a present alone. There are $n(S_2(n,k-1)$ partitions of this type since removing present $a$ and the present that is left alone gives us a valid partition of $n$ presents in $k-1$ groups of at least two presents each, and there are $n$ possibilities for the present that could have been with $a$.
Finally I would like to say  there is a similar recurrence for $S_r(n,k)$ and that the numbers $S_2(n,k)$ are also called the Ward numbers and they are the coefficients of the Mahler Polynomials.
