Why is it important that the product of two polynomials have only whole number exponents?

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Combinations allow 10,000, permutations allow 5,040.

Step-by-step explanation:

If we allow for repetition, we can have 10^4 possible combinations, if we can start numbers with 0. This is a base 10 system with 4 different objects.

If we don't allow for repetition, we will have 10 possible objects and a subset of 4. Every time we use a digit, we cannot reuse it, so we first have 10 possible digits for the 1st digit.

We have 10*9*8*7 possible combinations if we were to fill in the rest.

0XXX

Since we chose 0, we have 9 other digits remaining (9*8*7 possible permutations)

01XX

8 other digits. (8*7 possible permutations in this system)

018X

7 digits are possible now. (7 possible permutations if we pick 1 more digit)

In conclusion, we would have 10*9*8*7 (10!/6! or 5040) possible permutations in this system. Compared to using combinations where order and reuse is allowed, we have 10^4 possible combinations, or 10000. Combinations win.