Problem 46054. Count trailing zeros in a primorial

ChrisR

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Cody Problem 44068 asked us to count the trailing zeros in a factorial. This problem deals with the primorial. If  is the nth prime number, then the primorial # is the product of the prime numbers up to and including . For example, if n = 5, then # =  = 2310, which has one trailing zero.
Count the trailing zeros in the primorial #. Because primorials become large quickly, for large n it helps to derive a formula for the number of trailing zeros.*
Please make your code general because I might add more tests later.
See also Cody Problems 46057 "Find the last non-zero digit in a primorial" and 46060 "Identify primorial primes".
*;-)