Implement decay simulation and trailing-zero counting

Implement the following in Python:

1. Radioactive decay simulation: Half-life is 1 day. Write a simulation function that takes:

   • input: integer m (number of days), integer n (number of atoms)
   • process: simulate independent decay over time (you may model survival each day with probability 0.5, or equivalently sample each atom’s decay time)
   • output: the final state of each particle (e.g., a boolean list/array of length n indicating survived/decayed after m days) and/or the number of survivors. Also discuss runtime and how you would validate correctness against the analytic Binomial(n, 0.5^m) result.

2. Factorial and trailing zeros:

   • (a) Implement a factorial function in Python (mention multiple approaches).
   • (b) Given an integer n, compute the number of trailing zeros in n! efficiently (better than computing n! explicitly).