AirPort Extreme Roaming Network
In 2020 - get a Dream Machine if you can. The software does all this automagically. Then add AP as needed to complete your coverage based on your measured quality and channel needs.

Your question is probably too broad, but a pair of recent 802.11ac AirPort extremes might actually run well with 50 clients. I would get a third for 75+ or if your clients aren't on 802.11 AC substitute 2 Express for each one Extreme to get adequate bandwidth over WiFi.

Don't place them all together on one table. The closer they are to the external walls of the room, the better your interference situation will be.
Do link them with cat 5e or better ethernet cable
The one connected to the internet is the router and you'll have plenty of reservations out of the box.
Set the rest up in bridge mode.
Run a wireless scan using airport utility on iOS or get a Mac program like WiFi Explorer so you can assign channels manually and troubleshoot things like spectrum noise to signal and channel overlap and utilization.
If all of your clients are Mac and iOS, and if you place the base stations equally around the space,  toggling Wi-Fi off and on on the clients should do an acceptable job of balancing the load and matching each client to the closest base station
I would probably make different SSID for both 2.4 and 5 GHz and for each base station. You can make the password the same for all of them in that way if you need to go in and read distribute things you can.

If you set things up as I have described, you can choose and arbitrarily large pool of IP addresses, devices will retain the same IP address in general, UDP broadcast won't be a problem.

Set up the first Airport Extreme with Channel 11 and Channel 36
The rest go in bridge mode:

channel 6 and 153
channel 1 and 161
channel 11 and 157 (note 11 overlaps with the first - so make sure this is quite far from the first base station)

Avoid channel 149 since it's used initially for AirDrop/AirPlay. Don't use any channels other than 1, 6 and 11 on 2.4 GHz. If you have more complicated wifi planning needs, get a tool like wifi explorer to actually measure your network conditions in real time.