If $a, b$ and $c$ are positive real numbers, then

Match List-I with List-II

List-I	List-II
(Expression)	(The Least value of the expression)
(A) $(a + b)(b + c)(c + a)$	(I) $8abc$
(B) $(a + b + c)(ab + bc + ca)$	(II) $9a^2b^2c^2$
(C) $(a^2b + b^2c + c^2a)(ab^2 + bc^2 + ca^2)$	(III) $9abc$
(D) $(a + b)^2(b + c)^2(c + a)^2$	(IV) $64a^2b^2c^2$

Choose the correct answer from the options given below:

For expression $(a + b)(b + c)(c + a)$, apply AM-GM to each bracket:

$a + b \geq 2\sqrt{ab}$

$b + c \geq 2\sqrt{bc}$

$c + a \geq 2\sqrt{ca}$

Multiplying all three inequalities:

$(a + b)(b + c)(c + a) \geq 2\sqrt{ab} \times 2\sqrt{bc} \times 2\sqrt{ca}$

$= 8\sqrt{ab \times bc \times ca}$

$= 8\sqrt{a^2b^2c^2}$

$= 8abc$

Equality holds when $a = b = c$.

Minimum value $= 8abc$

(A) matches with (I)

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For expression $(a + b + c)(ab + bc + ca)$, let $a = b = c = k$:

$(a + b + c)(ab + bc + ca) = (3k)(3k^2)$

$= 9k^3$

$= 9abc$

Minimum value $= 9abc$

(B) matches with (III)

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For expression $(a^2b + b^2c + c^2a)(ab^2 + bc^2 + ca^2)$, let $a = b = c = k$:

First bracket: $a^2b + b^2c + c^2a = k^3 + k^3 + k^3 = 3k^3$

Second bracket: $ab^2 + bc^2 + ca^2 = k^3 + k^3 + k^3 = 3k^3$

Product: $(3k^3)(3k^3) = 9k^6$

$= 9(k^3)^2$

$= 9(abc)^2$

$= 9a^2b^2c^2$

Minimum value $= 9a^2b^2c^2$

(C) matches with (II)

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For expression $(a + b)^2(b + c)^2(c + a)^2$, this is the square of expression (A):

From (A): $(a + b)(b + c)(c + a) \geq 8abc$

$[(a + b)(b + c)(c + a)]^2 \geq (8abc)^2$

$(a + b)^2(b + c)^2(c + a)^2 \geq 64a^2b^2c^2$

Minimum value $= 64a^2b^2c^2$

(D) matches with (IV)

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Final matching:

(A) → (I) : $8abc$

(B) → (III) : $9abc$

(C) → (II) : $9a^2b^2c^2$

(D) → (IV) : $64a^2b^2c^2$