The following question is asked in hashing
There are 100000 buckets and each bucket has a capacity of 10. A bucket may not have the same item more than once. With random allocation and replacement, if there are 1000 items, how many items will be in at least 100 buckets?
Additionally, I want to find the answer to 100 buckets and less than 100 buckets. From above I know there are 1000 distinguishable items and 100000 containers and so $100000^{1000}$ ways.
The following was my attempt for at least 100 buckets.
I thought of using the pigeon-hole principle where because of replacement the same set of 10 items can be picked for each bucket. So 10 distinguishable items would be in at least 100 buckets. Would 10 also be the answer for 100 buckets and less than 100 buckets? What threshold in my case of 100 buckets would this not apply?
PostScript- Mention any problem-solving resources for combinatorics.
Sources - Probability that at least K cards will go into a bucket https://chrispiech.github.io/probabilityForComputerScientists/en/part1/combinatorics/ https://www.johndcook.com/blog/select_with_replacement/
