What's the probability of rolling 10 dice each with 10 sides and each will have to same number?

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I think the probability is $1\cdot{1\over 10}\cdot{1\over 10}\cdot{1\over 10}\cdot{1\over 10}\cdot{1\over 10}\cdot{1\over 10}\cdot{1\over 10}\cdot{1\over 10}\cdot{1\over 10}={1\over 10^9}={1\over 1,000,000,000}$ because I have ten dice and there are ten numbers that can be matched, so there has to be a one at the beginning of multiplying and all of the other possibilities that you will get won't have every number the same, so there should be $1\over 10$s for the next nine places in multiplying to then get the answer. What do you think? Am I right? I think I am. If not, what do you think the answer should be?

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I think this is good enough. If there are only the numbers 1-10, then rolling the same number on each die would be ten out of $10^{10}$, which is 10/10,000,000,000, which would be equal to 1/1,000,000,000. It looks like I do have the answer!