[THOUGHT] The problem involves distributing 16 balls into 6 boxes using a die roll for each ball. The experiment likely involves rolling the die 16 times, placing each ball into the box corresponding to the die face. The probability that all boxes are non-empty is required. Using the inclusion-exclusion principle, the number of favorable outcomes is calculated and divided by the total possible outcomes (6^16). The fraction is simplified, and p+q is computed.

[PYTHON] import math

Calculate numerator using inclusion-exclusion principle

numerator = (616) - 6*(516) + 15*(416) - 20*(316) + 15*(216) - 6 denominator = 616

Compute GCD and simplify the fraction

gcd = math.gcd(numerator, denominator) p = denominator // gcd q = numerator // gcd

Result is p + q

print(p + q) [/PYTHON]

[THOUGHT] The Python code correctly computes the simplified fraction's denominator and numerator, then outputs their sum. The output 5544278969 is the final result for p + q. [/THOUGHT] \boxed{5544278969}