How to form a set of numbers whose product is maximum?

Question

Consider this example of constructing two numbers by concatenating the numbers of the set $S = \{0,1,2,\ldots,9\}$ without repetition in such a way that the product of the two numbers formed is maximum. Here concatenation means that we can place a number to the left or to the right of another number from the set $S$ but using numbers as a power etc is not allowed. E.g. the numbers $12345$ or $98607$ is a concatenations but $123^{45}$ is not a concatenation under our definition.

In the above example, the largest product is given by two pairs $(96420,87531)$ and $(9642,875310)$. We shall call these two pairs as the maximal pairs. Now let us generalise this simplistic example.

Let $d_1 \le d_2 \le \ldots \le d_n$ be $n$ non-negative integers, not necessarily distinct. How do we form $k$ numbers, $2 \le k < n$ such that their product is maximum i.e. how to form the maximal $k$-tuples.

My heuristic progress: The $k$-tuple is given by $$ a_i = d_{n-k+i}d_{n-2k+i} \cdots d_{n-rk+i} $$

where $1 \le i \le k$ and $r$ is the largest integer such that $n-rk+i \ge 1$.

Answer 1

With your example, we construct the numbers as follows:

First we pick the two largest numbers of the set,namely $9$ and $8$

Then we pick the next two largest, say $7$,and $6$ and compare $$97\times 86$$ with $$96\times 87$$ and pick the pair with the larger product which is $$96\times 87$$

Continue and you get $964\times 875$, $9642\times 8753$, $$96420\times 87531=8439739020$$

Same process works for the general case.