What is the largest sum that results when one of the arrangements of the digits 2017 is added to one of the arrangements of the digits of 2016, if none of the digitsv0, 1, or 2 can occupy the same position in both numbers?
Since we want to maximize the sum, we want to maximize each place value of the numbers. The largest digit that can go in the thousands place is 2, so 2012 will be the number with the largest digits. Since we want to maximize the sum, we want the largest digit of the other number to go in the tens place. The largest digit that can go in the tens place is also 2, so 2012 still has the largest digits. Similarly, we place 1 in both of the units places. The number formed by the arrangement of 2017 is 2017, and the number formed by the arrangement of 2016 is 2016. Adding these numbers, we get $2012 + 2017 = \boxed{4029}$.
To find the largest sum that results when the arrangements of the digits 2017 and 2016 are added, we need to find the largest arrangements of these numbers and then add them together.
Step 1: Find the largest arrangement of the digits of 2017.
Since none of the digits 0, 1, or 2 can occupy the same position in both numbers, the largest arrangement of the digits of 2017 can be obtained by arranging the remaining digits (7 and 1) in descending order. Therefore, the largest arrangement of the digits of 2017 is 771.
Step 2: Find the largest arrangement of the digits of 2016.
Similarly, to find the largest arrangement of the digits of 2016, we need to arrange the remaining digits (6 and 0) in descending order. Therefore, the largest arrangement of the digits of 2016 is 660.
Step 3: Add the two arrangements together.
Adding 771 and 660, we get:
771
+ 660
______
1431
Therefore, the largest sum that results when one of the arrangements of the digits 2017 is added to one of the arrangements of the digits of 2016, with the given conditions, is 1431.