How do I specify the polygons inside a circle.
I mean the polygons inside the circle completely and also the polygons that partly are inside the circles.
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If they are in separate feature classes, you can spatially join the polygons to the circle. The resultant file will contain attributes of both. You can then summarize by a unique value in your circles feature class, and get the resultant count of those instances.
Assuming that the circle and polygons are in separate feature classes, you could use "Select by location" to select the polygons that intersect the circle. From there, you could export the selected polygons, or do whatever you need to do to them.