If the product of $n$ positive numbers is $n^n$, then what is the minimum value of their average for $n = 6$?

6

The correct answer is Option (3) → 6

Given: Product of $n$ positive numbers is $n^n$

For $n = 6$, product = $6^6 = 46656$

Let the numbers be $x_1, x_2, ..., x_6$ such that:

$x_1 x_2 x_3 x_4 x_5 x_6 = 6^6$

By AM ≥ GM inequality:

$\frac{x_1 + x_2 + x_3 + x_4 + x_5 + x_6}{6} \ge \sqrt[6]{x_1 x_2 x_3 x_4 x_5 x_6}$

$\Rightarrow \text{Average} \ge \sqrt[6]{6^6} = 6$

Equality holds when $x_1 = x_2 = ... = x_6$

So, minimum average = 6