imc

Let's first find how many possibilities there would be if they were all distinguishable, then divide out the ones we overcounted. There are $7!$ ways to order $7$ objects. However, since there's $3!=6$ ways to switch the yellow tiles around without changing anything (since they're indistinguishable) and $2!=2$ ways to order the green tiles, we have to divide out these possibilities: $\frac{7!}{3!\cdot 2}=\boxed{420}$.