[THOUGHT] Given $N \sim \mathrm{Binomial}(20, 1/2)$, we need to find $\mathbb{E}[N^4]$. The fourth moment of a Binomial random variable can be calculated using the formula for the central moments and then converting to raw moments. For a Binomial random variable $N \sim \mathrm{Binomial}(n, p)$, the moments are given by:

Plugging in $n = 20$ and $p = \frac{1}{2}$:

Finally, we need to calculate $K = 1000 \cdot \mathbb{E}[N^4]$:

Let's verify this using Python.

[/THOUGHT]

[PYTHON] import sympy as sp

Define the parameters for the Binomial distribution

n = 20 p = 1/2

Define the symbol for N

N = sp.symbols('N')

Define the Binomial distribution

binom_dist = sp.stats.Binomial('N', n, p)

Calculate the fourth moment E[N^4]

fourth_moment = sp.stats.E(N**4, binom_dist)

Calculate K = 1000 * E[N^4]

K = 1000 * fourth_moment

Print the result

print(int(K))

[THOUGHT] Given the error, let me manually compute the fourth moment using the correct formula or use an alternative approach. We already derived it as 13642.5. Therefore, $K = 1000 \times 13642.5 = 13642500$. [/THOUGHT] \boxed{13642500}