Can we identify the largest product of two numbers made from four given digits

Question

My problem is, given 4 positive integers (single digits 0-9), is there a generic way to identify which combination of them will yield the largest product of a single multiplication?

So to clarify if my digits were: 3, 8, 1, 6

I think the largest number I could make given a single multiplication would be 8 * 631 = 5048

or in generic terms, the largest single digit multiplied by a combination of the rest of the digits in a sequence of decreasing magnitude.

1. is that correct?
2. is there a way to formally state that? Sort of a proof.

Answer 1

It should be obvious that the digits in each number must be decreasing. It looks like you prohibit $abcd (\times 1)$. We can sort the digits $a \ge b \ge c \ge d$ - do you allow pairs to match? First let's look at splitting the digits $3$ and $1$. We need to compare $a \times bcd, b \times acd, c \times abd, d \times abc$. Comparing $a \times bcd=100ab+10ac+ad$ with $b \times acd=100ab+10bc+bd$ we see the former is larger because $a \ge b$ A similar argument will show that $a \times bcd$ beats the others.

Similarly we can compare $ac\times bd=100ab+10(ad+bc)+bd$ with $ad \times bc=100ab+10(ac+bd)+cd$. We would rather have more copies of the big $a$, so the second wins.

Finally we need to compare $a \times bcd=100ab+10ac+ad$ with $ad \times bc=100ab+10(ac+bd)+cd$. The second is the winner.