Jelly, 6 bytes

!ÆEṫ3Ḣ

Uses the counterproductive approach of finding the factorial then factorising it again, checking for the exponent of 5 in the prime factorisation.

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!              Factorial
 ÆE            List of prime exponents, e.g. 120 -> [3, 1, 1]
   ṫ3          Remove the first two elements - if n < 5, this results in an empty list
     Ḣ         Pop first, returning the exponent of 5 if ÆE returned a list of length >= 3,
               otherwise 0 is returned implicitly returned instead

Note that we can't index with 3ị since ị works cyclically, and for n = 4 it would extract 3 from the factorisation list [3, 1].

Jelly, 5 bytes

!Æfċ5

Uses the counterproductive approach of finding the factorial then factorising it again, checking for the exponent of 5 in the prime factorisation.

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!              Factorial
 Æf            List of prime factors, e.g. 120 -> [2, 2, 2, 3, 5]
   ċ5          Count number of 5s