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Only five words put D in the second box and M in the third, and all of them start with A.
ADMIN, ADMIT, ADMAN, ADMIX and ADMEN are the whole list. The pattern is really the AD prefix followed by an M, and English does not build a five letter word this way from any other opening.
Those two turn up far more than the rest put together. ADMAN and ADMEN are a singular and plural pair, and ADMIX is a technical verb meaning to mix something in. Three words is effectively the whole useful list.
Yes, it stands on its own now as a shortening of administration or administrator. It is the most used word in this group.
ADMAN is one person working in advertising and ADMEN is the plural. They are the only singular and plural pair here.
A D in the second box already narrows things sharply, and following it with M leaves almost nothing standing. Only the AD prefix survives.