Count Worksheets in multiple files

I need to build a macro that can read through a list of file names and return the number of worksheets that are present in each of the file (this is an audit mechanism to ensure that the correct number of worksheets are present in a series of files created via another process). The macro would need to establish the path of the folder where the files are located (all in the same folder), then locate the first file, identify the worksheet count, and repeat for the next file etc. For example the file requiring this functionality looks like:

wkshtcnt.PNG

I thought I could do this with a formula by simply referencing the file names but I believe Excel does not have a straightforward formula for worksheet counts.
Thanks!

Are the files going to be open when you run this code, and if not is it okay if the code opens them, counts the sheets, and then closes them?

The files would not be open. It would be ok if the code opens and closes them. I will add and additional piece of context in case this changes your thinking on this, the number of files will be large...over 2,000....
Thank you for your help

Preferably have the user select the folder (the user will probably be me so not overly concerned about "user friendliness" but upon each use of this the target files will be in a different folder).

None of these files will be password protected.

Thank you!

Thank you Leith Ross!
I made the necessary adjustments to orient your code to my file and this works beautifully! One thing I noticed is that the returned worksheet count is always 1 more than actually exists in the file (is it possible that the tab that appears as a default in excel thatyou click to add a new tab being counted?) I manually added worksheets to check this and it is always +1. I can work around this but thought I would ask if you can think of an adjustment for this. Just so you know, the only change I made to your code (other than the location references) was to remove the 'n=n+1' statement in the ' Copy the count to the cell one column to right of the workbook name in the list' section - the original code recorded the results in every other row as opposed to consecutive rows. I only point this out in case this has anything to do with the +1 worksheet count issue that I mention.
Thanks for your help.

Hi Leith Ross, this is perfect! Thank you very much for your expertise and help!