Solution:

If you know the formula for picking a subset of size k from a set of size n, with replacements allowed, C(n+k-1, k) , it is clear that there is exactly one order for every set of 6 values. There are 6 values, and we are picking 6, so the answer is 11 choose 6 = 462.

If you don't know this, we can look at it differently: The graph on the left is another representation of the picture in the problem statement. Each column contains a "bar" with height ranging from 1 to 6, representing the value of each die. You will see that the bars get progressively higher. They cannot go down; otherwise some die has a smaller value than one before it. We can trace the tops and lefts of the bars in blue, and notice a path that forms. As a matter of fact, every path going from (0,1) to (6,6) using only ups and rights will represent a unique result which works. There are 5 ups and 6 rights, and C(11, 6) = 462 ways to order them. (We just select 6 to be rights, and let the rest be ups)