The value of ` ((1)/( ( p - n)( n - q)) + (1)/( (n - q) ( q - p)) + (1)/(( q - p) ( p - n)))` is

Options

A. 1
B. 0
C. `p + q + n`
D. `(2 n)/(p + q)`

Answer

To solve the expression

\[ \frac{1}{(p - n)(n - q)} + \frac{1}{(n - q)(q - p)} + \frac{1}{(q - p)(p - n)}, \]

we will find a common denominator and simplify the expression step by step.

### Step 1: Identify the common denominator

The common denominator for the three fractions is

\[ (p - n)(n - q)(q - p). \]

### Step 2: Rewrite each term with the common denominator

We can rewrite each term as follows:

1. The first term becomes:
\[ \frac{(q - p)}{(p - n)(n - q)(q - p)}. \]

2. The second term becomes:
\[ \frac{(p - n)}{(n - q)(q - p)(p - n)}. \]

3. The third term becomes:
\[ \frac{(n - q)}{(q - p)(p - n)(n - q)}. \]

### Step 3: Combine the fractions

Now we can combine the fractions:
\[ \frac{(q - p) + (p - n) + (n - q)}{(p - n)(n - q)(q - p)}. \]

### Step 4: Simplify the numerator

Now, let's simplify the numerator:
\[ (q - p) + (p - n) + (n - q). \]

When we combine these terms, we see that:
- \(q\) and \(-q\) cancel out,
- \(p\) and \(-p\) cancel out,
- \(n\) and \(-n\) cancel out.

Thus, we are left with:
\[ 0. \]

### Step 5: Write the final result

Now substituting back into our combined fraction, we have:
\[ \frac{0}{(p - n)(n - q)(q - p)} = 0. \]

Therefore, the value of the given expression is

\[ \boxed{0}. \]