Variation argument is everywhere

Today I am asked a high school level question: given $1 \geq p_1 \geq p_2 \cdots p_n \geq 0$ and find a subset of these number such that to maximize the quantity

$$F(S) = \sum_{i=1}^{|S|}\frac{p_{\alpha_i}}{1 - p_{\alpha_i}} \left( \prod_{j=1}^{|S|}(1-p_{\alpha_j})\right).$$
If one would like to use the search naively, the complexity will be of course large as $n!$. I want to say a simple variational argument is although simple but useful. But supposing adding one more element, deleting one element, or replacing one element, one can see quickly the description for this optimal subset is $\tau$ that
$$S := \{p_i\}_{1 \leq i \leq \tau}, \tau := \min \left\{n \in \mathbb{N}: \sum_{i=1}^\tau \frac{p_i}{1-p_i} \geq 1\right\}.$$
Thus the complexity is reduced to $O(n)$. So let us always think about the variational argument.