Suppose that n indistinguishable balls are to be arranged in N distinguishable boxes so that each distinguishable arrangement is equally likely. If $n \geq N$, show that the probability no box will be empty is given by
$\frac{(\begin{matrix}
n-1 \\
N-1 \\
\end{matrix})}{(\begin{matrix}
N+n-1 \\
N-1 \\
\end{matrix})}$.
I am supposed to use intuition and nothing more.
Best Answer
Hint: This uses a Stars and Bars argument, actually, two of them, Please see the Wikipedia article for details.
For the denominator, we want the number of solutions of the equation $$x_1+x_2+x_3+\cdots +x_N=n$$ in non-negative integers.
For the numerator, you want the number of solutions of the same equation in positive integers.
The Wikipedia article gives formulas both for the number of non-negative solutions and for the number of positive solutions. There is a reasonably thorough explanation of why the formulas are correct.