It

*really* would have helped if she'd provided her reasoning. As it is, her "work" is quite cryptic. My guess is that she's not showing a

*bunch* of her reasoning!

For the two-dice case, I

*think* that she's assuming that the first die has been thrown, so she knows the first value. To get a sum of eight, the first dice has to show more than 1 (because the highest value for the second die is 6, and 1 + 6 doesn't equal 8). So it'll be 2 through 6. For the second die, the value has to be 8 - (whatever the first value was) to equal 8, so she's needing a 2 through a 6. The "C(5, 1)" may be her way of saying that she's counting the number of ways to get one of those five values. Maybe...?

Then maybe for the second situation, she's saying that she's thrown the first die, and gotten something between 1 and 6. (Since she's throwing three dice, she can use 1's this time: 1 + 1 + 6 = 8.) Suppose the first die rolls a 1. Then she's looking at the number of ways of getting 7 with the remaining two dice. Say she rolls an "x" on the second die; then she's needing 7 - x for the third die. Since 1

< x

< 6 for the second die, then the target value for the third die is 6, 5, 4, 3, 2, or 1. There are six ways to pick one of these values.

Now suppose the first die rolls a 2. Then she's looking at the number of ways of getting 6 with the remaining two dice. Using "x" again for the value of the roll of the second die, she'll be needing 6 - x for the third die. The target value will be 5, 4, 3, 2, or 1. There are five ways to pick one of these values.

And so forth.

But that's just my guess.