Assume there are $a,b$ and $c$ numbers of songs in the first, second and third playlist respectively..
Therefore $a+b+c=n$. The number we can choose one song from each playlist is $abc$.
We have to find the maximum value of $abc$ corresponding to the condition $a+b+c=n$.
Let $L=abc-\lambda(a+b+c)$
Therefore $\frac{\partial L}{\partial a}=bc-\lambda, \frac{\partial L}{\partial b}=ac-\lambda, \frac{\partial L}{\partial c}=ab-\lambda$
For $\frac{\partial L}{\partial a}=\frac{\partial L}{\partial b}=\frac{\partial L}{\partial c}=0$ we get $ab=bc=cd=\lambda$ and $a=b=c$
Therefore $a^2b^2c^2=\lambda^3 \Rightarrow a^6=\lambda^3\Rightarrow a=\sqrt{\lambda}=b=c$
By the given condition $a+b+c=n\Rightarrow 3\sqrt{\lambda}=n\Rightarrow \sqrt{\lambda}=\frac{n}{3}$
Therefore the maximum value of $abc$ is $({\sqrt{\lambda}})^3=\frac{n^3}{27}$
Therefore the number of ways of choosing three songs consisting of one song from each play list is
- $\leq\frac{n^3}{27}$ for all $n$