The probability of the sum of $10$ dice rolls adding up to $57$

Question

So the question is: given that you roll $10$ dice, what is the probability of the sum of the total dice rolls adding up to $57$?

I know that there are three ways to do this:

1. Seven die rolls must be $6$ with three $5$s
2. Eight die rolls must be $6$, one die roll must be $5$ and one must be $4$
3. Nine die rolls must be $6$, and one roll must be $3$

The solution states that the probability of the events are:

1. $ \binom{10}{3} \cdot \frac{1}{6^{10}}$
2. $ \binom{10}{1} \cdot \binom{9}{1} \cdot \frac{1}{6^{10}} $
3. $ 10 \cdot 9 \cdot \frac{1}{6^{10}} $

I really don't understand why the probabilities work this way. I would really, really appreciate it if someone could perhaps explain this in a more intuitive way for me.

Edit: I am really sorry for the mistake. Edited so that the question reads sum up to 57.

Edit 2: Also, I think my solution sheet is missing the fact that you should sum all of these probabilities and set them over $6^{10}$. I apologize for the mess and I appreciate all the comments that pointed this out.

Answer 1

For any of the three solutions, for each die, there is a $\tfrac{1}{6}$ chance of rolling the right number, so there is a $\tfrac{1}{6^{10}}$ chance of that specific combination turning up.
But it doesn't matter if the first die rolled a $6$ and the last the $3$ you needed, or the other way around. So we're going to try to figure out how many possible ways there are of rolling any of those combinations.

1. Seven $6$s and three $5$s
   There is a total of $\binom{10}{3} = 120$ ways to roll this combination, each of which has a probability of $\tfrac{1}{6^{10}}$, so the total probability is $\binom{10}{3} \cdot \frac{1}{6^{10}} = \frac{120}{6^{10}}$
2. Eight $6$s, a $5$, and a $4$
   This one isn't that hard. We can roll this combination in a total of $\binom{10}{1} \cdot \binom{9}{1} = 10 \cdot 9$ ways, which you can verify for yourself: there are $10$ dice that can be the $5$, after which there are $9$ dice that can be the $4$, with the rest being $6$s.
   Since each has a probability of $\tfrac{1}{6^{10}}$, the total probability here is $\binom{10}{1} \cdot \binom{9}{1} \cdot \frac{1}{6^{10}} = \frac{90}{6^{10}}$
3. Nine $6$s and a $3$
   This one is the simplest of the three. There are just $\binom{10}{1} = 10$ ways of rolling this combination. Again, you can verify this yourself; any one of the dice can be $3$ while all others are $6$. The total probability here is $\frac{10}{6^{10}}$

Now that we have the separate probabilities, all that's left to do is add them up.

$\begin{align} P(57) & = \frac{120}{6^{10}} + \frac{90}{6^{10}} + \frac{10}{6^{10}} \\ & = \frac{220}{6^{10}} \\ & \approx 3.638\times10^{-6} \end{align}$

Note that this means that out of the $6^{10}$ possible rolls of the dice, only $220$ will sum up to $57$.