Using SUMPRODUCT instead of COUNTIFS

I needed to introduce an OR-type of statement into my countifs, but I've read that you can't do that, that you need to use SUMPRODUCT instead.

I'm counting injuries per team per type per week. I'm counting injuries, so I'm looking for: How many people on this TEAM had this INJURY TYPE on any of these 4 DATES.

I've attached a manually entered .xls to show my expected outcome.

In my work document, I have multiple tabs (Week 1, Week 2, etc.) each of which have a manually entered date range. I'd like to be able to copy this array of formulas into each tab so I don't have to adjust the formulas each time. They'll just adjust with the manually entered date ranges.