At an event , If there are m actors, n singers and r tables , find the number of ways you can place the guests. (tables are alike)

Question

So the question asks us to find the total arrangements possible given the no. of singers, actors and tables given the fact that -

• A table can have either only singers or only actors, not both.

• No table can have only one actor

• None of the tables are left empty

Here's what I tried -

Select m tables for the actors (mCr) , and the remaining r-m tables for n singers (m-nCr).

The no. of ways to distribute distinct objects into alike boxes is S(n,r). (Stirling no. of second kind)

Therefore - S(m,m) x S(m-r,n) is what I am left with.

How do I make sure none of the actors are seated alone?

Answer 1

Hint$_1$: You want to use the Associated Stirling numbers of the second kind.

Hint$_2$: For this particular case, you can try using inclusion-exclusion. Let the sets $A_i = \{ \text{distribution of $m$ actors in $k$ tables such that the $i$-th actor is alone}\}$. Show that $|A_i|=S(m-1,k-1)$. What is then $\left |\bigcap _{i\in I}A_i\right |$?

Compute $S(m,k)-\left |\bigcup _{i=1}^m A_i\right |$.