Let $$x_1, x_2, x_3, x_4$$ be in a geometric progression. If 2, 7, 9, 5 are subtracted respectively from $$x_1, x_2, x_3, x_4$$ then the resulting numbers are in an arithmetic progression. Then the value of $$\frac{1}{24}(x_1 x_2 x_3 x_4)$$ is :

Let $$x_1 = a$$, $$x_2 = ar$$, $$x_3 = ar^2$$, $$x_4 = ar^3$$ be in geometric progression.

After subtracting 2, 7, 9, 5 respectively, the resulting numbers $$a-2, ar-7, ar^2-9, ar^3-5$$ are in arithmetic progression.

For an AP, consecutive differences are equal:

$$(ar - 7) - (a - 2) = (ar^2 - 9) - (ar - 7)$$

$$a(r-1) - 5 = a(r^2-r) - 2$$

$$a(r - 1 - r^2 + r) = 3$$, which gives $$-a(r-1)^2 = 3$$ ... (i)

$$(ar^2 - 9) - (ar - 7) = (ar^3 - 5) - (ar^2 - 9)$$

$$a(r^2 - r) - 2 = a(r^3 - r^2) + 4$$

$$a(2r^2 - r - r^3) = 6$$, which gives $$-ar(r-1)^2 = 6$$ ... (ii)

Dividing (ii) by (i): $$r = 2$$.

Substituting back into (i): $$-a(1) = 3$$, so $$a = -3$$.

The GP terms are: $$x_1 = -3$$, $$x_2 = -6$$, $$x_3 = -12$$, $$x_4 = -24$$.

Verification: $$-5, -13, -21, -29$$ form an AP with common difference $$-8$$. ✓

$$x_1 x_2 x_3 x_4 = (-3)(-6)(-12)(-24) = 5184$$

$$\frac{1}{24}(x_1 x_2 x_3 x_4) = \frac{5184}{24} = 216$$

Hence, the correct answer is Option D.