Add p(p – q), q(q – r) and r(r – p).
(a) \[{{p}^{3}}+{{q}^{2}}+{{r}^{2}}-pq+qr-pr\]
(b) \[{{p}^{2}}+2{{q}^{2}}+{{r}^{2}}-pq-qr+pr\]
(c) \[{{p}^{2}}+{{q}^{2}}+{{r}^{2}}-pq-qr-pr\]
(d) \[{{p}^{3}}+{{q}^{2}}+{{r}^{2}}-pq-qr-pr\]

Hint: Firstly simplify the expression by multiplying the terms in the parentheses using the rule a (b + c) = ab + ac where a, b and c are real numbers. Now apply the property \[a\cdot a={{a}^{2}}\] to solve the expression. Then express the algebraic expression according to their decreasing degree of exponents to get the final answer.

Complete step by step solution:
Adding the terms p(p – q), q(q – r) and r(r – p) can be mathematically expressed as,
p(p – q) + q(q – r) + r(r – p)
We know multiplication of an algebraic expression can be done in the form a (b + c) = ab + ac where a, b and c are real numbers.
Applying the above mentioned method to simplify the given expression we get,
pp – pq + qq – qr + rr – rp
We know the property \[a\cdot a={{a}^{2}}\] where a is a real number.
Applying the above mentioned property we get,
\[{{p}^{2}}-pq+{{q}^{2}}-qr+{{r}^{2}}-rp\]
By arranging the above expression according to the decreasing degree of the exponents we get,
\[{{p}^{2}}+{{q}^{2}}+{{r}^{2}}-pq-qr-rp\]
We get the simplified expression as \[{{p}^{2}}+{{q}^{2}}+{{r}^{2}}-pq-qr-rp\].
Hence option (c) is the correct answer.

Note: The possible error that you may encounter can be the incorrect use of operations. The rearranging of operation signs should be done carefully. Multiplying same algebraic terms should be done in accordance with the property \[a\cdot a={{a}^{2}}\] where a is a real number.