Set the profit equation to zero.

The profit $P$ is given by $P = n^2 - 10n + 24$.
For zero profit (break-even point), we set $P = 0$:

$$n^2 - 10n + 24 = 0$$

We need to find two numbers that multiply to 24 and add up to $-10$. These numbers are $-4$ and $-6$.
So, the equation can be factored as:

$$(n - 4)(n - 6) = 0$$

Set each factor equal to zero:

$$n - 4 = 0 \quad \text{or} \quad n - 6 = 0$$ $$n = 4 \quad \text{or} \quad n = 6$$

The shopkeeper must sell either 4 items or 6 items to make zero profit.

The number of items he must sell to make zero profit is $\text{4 or 6 items}$.