How will you power the router?
If possible, I would recommend keeping the ISP modem, router, and 16 port switch together in the study. It can make troubleshooting easier if the main components are together. It also allows you to have a UPS to battery backup the core of your network. (Very nice if you work from home)
If putting the wifi router if the study causes wifi issues in the family room, you could always add another wifi router in access point mode. Depending on how many wired ports you need in the family room, you may not need the 5 port switch if the access point is there.