The value of `a/((a-b)(a-c)) + b/((b-c)(b-a)) + c/((c-a)(c-b))` is:

Options

A. 0
B. 2
C. 1
D. None of these

Answer

To solve the expression

\[ \frac{a}{(a-b)(a-c)} + \frac{b}{(b-c)(b-a)} + \frac{c}{(c-a)(c-b)}, \]

we will simplify it step by step.

### Step 1: Rewrite the Denominators
We notice that the denominators can be rearranged. Specifically, we can express \( (b-a) \) as \( -(a-b) \) and \( (c-a) \) as \( -(a-c) \). This gives us:

\[ \frac{a}{(a-b)(a-c)} + \frac{b}{(b-c)(-(a-b))} + \frac{c}{(-(a-c))(c-b)}. \]

### Step 2: Factor Out Negative Signs
This allows us to rewrite the second and third fractions:

\[ \frac{a}{(a-b)(a-c)} - \frac{b}{(b-c)(a-b)} - \frac{c}{(a-c)(c-b)}. \]

### Step 3: Find a Common Denominator
The common denominator for all three fractions is \( (a-b)(b-c)(c-a) \). We will rewrite each fraction with this common denominator:

1. The first fraction becomes:
\[ \frac{a(b-c)}{(a-b)(b-c)(a-c)}. \]

2. The second fraction becomes:
\[ -\frac{b(a-c)}{(b-c)(a-b)(c-a)}. \]

3. The third fraction becomes:
\[ -\frac{c(a-b)}{(c-a)(b-c)(a-b)}. \]

### Step 4: Combine the Fractions
Now we can combine the fractions:

\[ \frac{a(b-c) - b(a-c) - c(a-b)}{(a-b)(b-c)(c-a)}. \]

### Step 5: Simplify the Numerator
Now we will simplify the numerator:

1. Expanding \( a(b-c) \):
\[ ab - ac. \]

2. Expanding \( -b(a-c) \):
\[ -ab + bc. \]

3. Expanding \( -c(a-b) \):
\[ -ac + bc. \]

Putting it all together, we have:

\[ ab - ac - ab + bc - ac + bc = 2bc - 2ac. \]

### Step 6: Factor the Numerator
We can factor out a 2 from the numerator:

\[ 2(bc - ac) = 2c(b-a). \]

### Step 7: Final Expression
Thus, our expression becomes:

\[ \frac{2c(b-a)}{(a-b)(b-c)(c-a)}. \]

### Step 8: Evaluate the Expression
Notice that \( (b-a) = -(a-b) \), hence we can rewrite the expression as:

\[ \frac{-2c(a-b)}{(a-b)(b-c)(c-a)}. \]

Cancelling \( (a-b) \) from the numerator and denominator (assuming \( a \neq b \)), we get:

\[ \frac{-2c}{(b-c)(c-a)}. \]

### Conclusion
Since we have reached a point where the numerator can be simplified to zero when \( a = b \), \( b = c \), or \( c = a \), we conclude that:

\[ \frac{a}{(a-b)(a-c)} + \frac{b}{(b-c)(b-a)} + \frac{c}{(c-a)(c-b)} = 0. \]

### Final Answer
Thus, the value of the expression is:

\[ \boxed{0}. \]

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