(a - b)$$^3$$​+ (b - c)$$^3$$​​+ (c -a)$$^3$$​​= ?, which of the following should come in place of ? so that relation always holds true?

Correct option is C

Given:

We are given the equation:

​$$(a - b)^3 + (b - c)^3 + (c - a)^3 = ?$$​​

Concept Used:

We can use the identity for the sum of cubes:

​$$x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$$​​

Solution:

In the given equation, let:

x = (a - b), y = (b - c), z = (c - a)

Now, applying the sum of cubes identity:

​$$(a - b)^3 + (b - c)^3 + (c - a)^3 - 3(a - b)(b - c)(c - a) = (a - b + b - c + c - a)((a - b)^2 + (b - c)^2 + (c - a)^2 - (a - b)(b - c) - (b - c)(c - a) - (c - a)(a - b))$$​​

Simplifying the first part of the right-hand side:

a - b + b - c + c - a = 0

Thus, the equation simplifies to:

0 × (expression) = 0 

​$$(a - b)^3 + (b - c)^3 + (c - a)^3 - 3(a - b)(b - c)(c - a) = 0$$ 

$$(a - b)^3 + (b - c)^3 + (c - a)^3 = 3(a - b)(b - c)(c - a)$$​​

So, the relation always holds true for the expression that equals 0.

Therefore, the correct choice is:

C.3 (a - b)(b - c)(c - a)