Answer
277200
Work Step by Step
We can use the general idea that we can order these coins, if they were distinguishable, in $12!$ ways. However, since they are indistinguishable, we can swap around the types of coins, so we divide by $4!*3!*2!*3!$ accounting for the pennies, nickels, dimes, and quarters respectively.
$$\frac{12!}{4!*3!*2!*3!}=\boxed{277200}$$