【Analysis】First find the number of denominators that have only the factors 2, 5, and both 2 and 5, and the conclusion can be drawn.
【Solution】In the 2015 fractions $\frac{1}{2}, \frac{1}{3}, \frac{1}{4} \cdots \frac{1}{2014}, \frac{1}{2015}, \frac{1}{2016}$, the denominators that have only the factor 2 are $2,4,8,16,32,64,128,256,512,1024$, a total of 10 numbers, and the denominators that have only the factor 5 are $5,25,125,625$, a total of 4 numbers,

The denominators that have both the factors 2 and 5 are $10,20,40,50,80,100,160, 200, 250, 320, 400, 500, 640, 800, 1000, 1250, 1280, 1600, 2000$, a total of 19 numbers,

So there are $10+4+19=33$ finite decimals in total,
Answer: There are 33 finite decimals.
10. (10 points) Convert the 2015 fractions $\frac{1}{2}, \frac{1}{3}, \frac{1}{4} \cdots \frac{1}{2014}, \frac{1}{2015}, \frac{1}{2016}$ into decimals. How many of them are finite decimals?