A round cake was cut with a knife $4$ times vertically in such a way that it is cut to maximum number of pieces.Find the number of ways of distributing these cakes among three people such that everyone gets at-least one piece.
My attempt: Maximum number of regions formed by $n$ straight lines in a plane is given by $1+\sum_{i=1}^n (i)$,so here the maximum cuts possible is $11$ now to me it seems that the rest is to count number of surjections between two sets of cardinality $11$ and $3$ respectively, which is $3! \times $StirlingS$2[11, 3]=3^{11} -3 \times 2^{11} + 3$.
But the answer says it is to be $3^{11} -3(2^{11}+1)$My instructor says that this is correct and he explains it as follows:
The total number of ways of distributing these cake pieces to $3$ people = $3^{11}$
This includes $3\times2^{11}$ ways of distributing the cake pieces in which one person will not get any cake pieces, and $3$ ways of distributing $11$ cake pieces in which only one person will get all the cake pieces.
Therefore,the required number of ways = $3^{11} -3(2^{11}+1)$.
But I am quite sure that I haven't committed any mistake in recognizing the model but I couldn't find a flaw in his reasoning either, also I couldn't convince myself why the counting surjections is not working here?!