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

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 $$n-rk+i \ge 1$$.

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 $$964\times 875$$, $$9642\times 8753$$, $$96420\times 87531=8439739020$$