Minor cleanup

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Mark 2023-10-04 09:29:46 -07:00
parent 10177fa934
commit 0744a88c00
Signed by: Mark
GPG Key ID: AD62BB059C2AAEE4

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@ -250,9 +250,10 @@ If we place $A$ and $B$ at opposing vertices, what is the effective resistance o
There are $\binom{n}{k}$ nodes in the $k^\text{th}$ layer. Each node in this layer has $k$ ones, so
there are $n - k$ ways to flip a zero to get to the $(k + 1)^\text{th}$ layer. In total, there are
$\binom{n}{k}(n - k)$ parellel connections from the $k^\text{th}$ layer to the $(k + 1)^\text{th}$
layer, creating an effective resistance of $(\binom{n}{k}(n - k))^{-1}$.
\vspace{2mm}
layer, creating an effective resistance of
$$
\frac{1}{\binom{n}{k}(n - k)}
$$
The total effective resistance is therefore
$$
@ -263,7 +264,7 @@ If we place $A$ and $B$ at opposing vertices, what is the effective resistance o
To calculate the limit as $n \rightarrow \infty$, note that
$$
\binom{n}{k}(n - k) = \frac{n!}{(n - k - 1)! \times k!} = n \binom{n-1}{k}.
\binom{n}{k}(n - k) = \frac{n!}{(n - k - 1)! \times k!} = n \binom{n-1}{k}
$$
So, the sum is
$$