I have no idea where to start with this exercise. Could please somebody help me to understand all the steps to solve this one?
One day n gentlemen came to the club. There are 2n identical pieces of cake available. Each gentlemen takes at least one and at most three pieces of cake. Assuming that all cakes are taken. In how many ways can this be arranged?
Assuming that all guests are indistinguishible.
Since everyone takes atleast $1$ cake, $n$ cakes are left. This is equivalent to finding solutions for the following equation.
$$x+2y=n$$
Each $y$ has a unique $x$, so it is sufficient to count $y$ that satisfy.
Finally, the total ways are
$$\lfloor \frac{n}{2} \rfloor$$