When Poisson Is Not Close Enough

You roll 24 fair six-sided dice and want the probability that at least four of them show a 5. A colleague replaces the Binomial count by a Poisson with the same mean:

from math import comb, exp, factorial

n, p = 24, 1 / 6
lam = n * p                                             # 4.0
approx = 1 - sum(exp(-lam) * lam**k / factorial(k) for k in range(4))
print(round(approx, 3))                                 # 0.567

That 0.567 is not the answer. Report the exact probability that at least four of the 24 dice show a 5.

Answer format: round to 3 decimal places.