CZ-SK-MO 2020 Problem 6

For each positive integer $k$ , denote by $P (k)$ the number of all positive integers $4k$ -digit numbers which can be composed of the digits $2, 0$ and which are divisible by the number $2 020$ . Prove the inequality $$P (k) \ge \binom{2k - 1}{k}^2$$ and determine all $k$ for which equality occurs.

(Note: A positive integer cannot begin with a digit of $0$ .)

(Jaromir Simsa)