Count ( applicants::anyFieldthatIsNeverEmpty ) can be defined in instiutions to count the number of applicants for a given institution.
If you define a "count of" summary field in applicants (must also count a field that is never empty), you can place that summary on your institution layout and you'll get the same count.
Or, you can set up a summary report with sub summary layout parts on a applicants based layout and you can also show the same counts for each institution.
This last option can list just the institutions or it can list the institution as a "sub head" with the individual applications listed grouped with it. It can also be a report that further filters the applicants being counted--such as all the applicants for a given year or from a specific part of the country.
What elements would the last option need to contain? Would they need to be in a portal?
No portal, the last option is a summary report with a specific sort order consistent with Sub summary layout parts on a layout that is viewed in List view.
This tutorial on summary reports may be helpful: Creating Filemaker Pro summary reports--Tutorial