Question

The value of $ \frac{1}{1 + p^{(y-z)} + p^{(x-z)}} + \frac{1}{1 + p^{(x-y)} + p^{(z-y)}} + \frac{1}{1 + p^{(y-x)} + p^{(z-x)}}, $ is:

• A. 3
• B. 1
• C. 0
• D. -1

Hint:

Look for symmetries and cyclic relationships in algebraic expressions to simplify them more easily.

Answer 1

The Correct Option is B

We are given the expression: \[ \frac{1}{1 + p^{(y-z)} + p^{(x-z)}} + \frac{1}{1 + p^{(x-y)} + p^{(z-y)}} + \frac{1}{1 + p^{(y-x)} + p^{(z-x)}}. \] Step 1: Notice that each fraction in the expression follows a similar structure with cyclic permutations of \( x \), \( y \), and \( z \). This symmetry suggests the possibility of a simplification where all terms might cancel out or equalize. Step 2: With the symmetry and recognizing the cyclic nature of the powers of \( p \), we see that when all terms are combined, they simplify to the value 1. This simplification arises from the symmetry in the exponents and the structure of the expression. Step 3: Therefore, the value of the given expression is \( \boxed{1} \).