[THOUGHT] I need to compute the fourth moment of a binomial random variable N ~ Binomial(20, 1/2). The fourth moment is E[N^4]. For a binomial random variable with parameters n and p, the moments can be computed using the moment generating function or by using factorial moments and polynomial expansion.

The fourth moment can be expressed in terms of the factorial moments: E[N^4] = E[N(N-1)(N-2)(N-3)] + 6E[N(N-1)(N-2)] + 7E[N(N-1)] + E[N]

For a binomial random variable: E[N] = np E[N(N-1)] = n(n-1)p^2 E[N(N-1)(N-2)] = n(n-1)(n-2)p^3 E[N(N-1)(N-2)(N-3)] = n(n-1)(n-2)(n-3)p^4

Given n=20, p=1/2, I can compute each term and sum them up to get E[N^4]. Then K = 1000 * E[N^4].

I'll use Python to compute this step by step to avoid calculation errors. [/THOUGHT] [PYTHON]

Given parameters

n = 20 p = 0.5

Compute factorial moments

E_N = n * p E_N2 = n * (n - 1) * p2 E_N3 = n * (n - 1) * (n - 2) * p3 E_N4 = n * (n - 1) * (n - 2) * (n - 3) * p**4

Compute fourth moment using factorial moments

E_N4_moment = E_N4 + 6 * E_N3 + 7 * E_N2 + E_N

Compute K

K = 1000 * E_N4_moment

Print the result as integer

print(int(K)) [/PYTHON]

[THOUGHT] The Python calculation confirms that K = 13072500. This is the exact integer value of 1000 times the fourth moment of a Binomial(20, 1/2) random variable. [/THOUGHT] \boxed{13072500}