In how many ways can $5$ different colored balls be placed into $8$ bins if no bin contains more than one ball?

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5 distinct different colored balls into 8 distinct bins and no bin contain more than one ball, how many ways can the balls be placed into the bins?

I got 8*7*6*5*4=6,720. But I am unsure if it is right. My thought process was that the first distinct ball can be placed in 8 bins, the second into 7 bins ect. and i used the product rule to multiply all the combinations together to get my answer. Any hints or help would be great thanks.