Question 49
There are $2019$ bags: each bag contains $2019$ beads with total weight $1$ kg. In each bag the beads are numbered by $1, 2, \ldots, 2019$. A proper collection is a collection of differently numbered beads containing at most one bead from each bag and having total weight not less than $1$ kg. Find the maximal possible value of $k$, if always there are at least $k$ different proper collections.