At the dashboard level there is currently no solution for that. Once your on the problems page you can filter it out as you see fit but you the functionality of pinning that with the filters to the dashboard is not available.
Very interesting use case tho. I would recommend tossing in a RFE: https://answers.dynatrace.com/content/idea/post.html?space=483
While this direct functionality is not possible, I would like to present a close alternative that is possible. This involves using the "entity lists" views (Transactions & Services, Hosts, Databases). You can filter these lists by "Problem Impact" and set it to Impacted, then pin this to your dashboard.
What you end up with is a breakdown of problems per entity type (which also supports the additional Management Zone filtering).
Of course this is not an exact number of Problems, but rather entities of each type that are part of Problems (so we may have 3 Services part of 1 Problem) but it is a popular alternative to achieve that breakdown of the impact. With the Problem tile next to it, it offers a very readable layout.
Please, also feel free to raise this as an RFE as Chad suggested earlier.
I hope this helps.