Probability - choosing an element from expanded form of $n\times n$ determinant

Question

An element from expanded form of $n\times n$ determinant is randomly chosen. Find the probability $p_n$ that randomly chosen element doesn't contain an element from the main diagonal. Find $\lim_{n\to 0}p_n!.$

Let $n=2$,

$$ \begin{vmatrix} a & b \\ c & d \\ \end{vmatrix}=ad-bc$$

We choose an element from expanded determinant form, $ad-bc$.

If we choose $ad$, then $p_n=0$.

If we choose $bc$, then $p_n=1$.

Let $n=3$,

$$\begin{vmatrix} a & b & c \\ d & e & f\\ g & h & i\\ \end{vmatrix}=aei-afh-bdi+bfg+cdh-ceg$$

If we choose $aei$, then $p_n=0$.

If we choose $afh$, then $p_n=2/3$.

If we choose $bdi$, then $p_n=2/3$.

If we choose $bfg$, then $p_n=1$.

If we choose $cdh$, then $p_n=1$.

If we choose $ceg$, then $p_n=2/3$.

Is this correct?

How can we find the closed form of $p_n$, for $n\times n$ determinant?

Answer 1

HINT: Prove by induction that each entry appears in the same number of terms ($(n-1)!$ terms for $n \times n$ matrix). Therefore the probability of choosing an element is same for all of them. Eventually there are $n$ entries in the main diagonal and $n^2$ in total, so the probability is...