The factors that are multiplied in the expansion of 30! are



Every factor of 15 of 30! is a pair of factors of 5 and 3.


These 5 multiples of 5, which are 5,10,15,20, and 30 all contribute
1 5 factor.  The factor 25 = 52 contributes 2 more factors of 5.  
So there are 7 factors of 5 in the prime factorization of 30!.

There are more than 7 factors of 3 (In fact there are 14 factors of 3). 

So 30! has 7 factors of 5. 

So the 7 factors of 5 can be paired together with 7 of the factors of 3,
making 7 factors of 15.

So the highest power of 15 that is a factor of 30! is 157, the seventh power.

Edwin