The given expression is:

1/(x^b * x^-c * 1) + 1/(x^c * x^-a * 1) + 1/(x^a * x^-b * 1)

To simplify this expression, we can use the properties of exponents.

1. Simplifying the first term:
1/(x^b * x^-c * 1) = 1/(x^(b-c))

2. Simplifying the second term:
1/(x^c * x^-a * 1) = 1/(x^(c-a))

3. Simplifying the third term:
1/(x^a * x^-b * 1) = 1/(x^(a-b))

Now, let's combine these simplified terms:

1/(x^(b-c)) + 1/(x^(c-a)) + 1/(x^(a-b))

To find the common denominator, we need to find the least common multiple (LCM) of (b-c), (c-a), and (a-b).

The LCM of (b-c), (c-a), and (a-b) will be equal to the absolute difference between the maximum and minimum values among (b, c, a).

Let's consider the possible cases for the values of a, b, and c:

Case 1: If a = b = c
In this case, the LCM will be 0, and the expression will be undefined.

Case 2: If a, b, and c are distinct positive integers
In this case, the LCM will be equal to the absolute difference between the maximum and minimum values among (b, c, a). Let's assume b > c > a. The LCM will be equal to (b - a).

Case 3: If a, b, and c are distinct negative integers
In this case, the LCM will also be equal to the absolute difference between the maximum and minimum values among (b, c, a). Let's assume b < c="" />< a.="" the="" lcm="" will="" be="" equal="" to="" (a="" -="" />

Case 4: If a, b, and c are a combination of positive and negative integers
In this case, the LCM will be equal to the absolute difference between the maximum and minimum values among (b, c, a). Let's assume b > c > a. The LCM will be equal to (b - a).

Therefore, no matter the values of a, b, and c, the LCM will always be equal to the absolute difference between the maximum and minimum values among (b, c, a).

So, the expression will reduce to:

1/(x^(b-c)) + 1/(x^(c-a)) + 1/(x^(a-b)) = 1/(x^(max(b, c, a) - min(b, c, a)))

Considering the given options:
A. 1
B. 0
C. -1
D. None

The correct option is D. None, as the expression does not reduce to a single term when a, b, and c are given. The expression will depend on the specific values of a, b, and c.