Selecting nearest 300 buildings (polygons/centroids) from list of points using ArcGIS

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I have been trying to look for an easy/quick way to select a certain amount of polygons (layer 1) from a 90 points shapefile (layer 2).

The idea is to identify the nearest 300 buildings from each of the 90 vaccination sites. Selecting these buildings based on a distance is easy but based on the number of features around it seems much more complicated.

I could not find any good way (I tried "Near", "Make Closest Facility", "Neighborhood location"...but they don't seem to do what I'm looking for).

I can't find any tool doing this. Doing this for only 1 point is ok but for a larger number of points is not obvious.

Any suggestion?

I'm using ArcGIS 10.3 for Desktop.



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