[nooonuii]

no. 18 is very easy if we use the Legendre formula. The solution is 24. Anyway, we can use elementary counting:

the number of zeros in this product depends on the number of 5 in the product because each 2 and 5 in the product produces one 0 but the number of 2 in the product is always more than the number of 5. So, it suffices to count the number of power of 5 in this product. Observe that there are 20 numbers in the sequence of number from 5 to 100 which is divisible by 5
so we have 20 5's, but 25, 50, 75, and 100 have another remaining 5 to count. Therefore, there are 24 5's in this product.