Bridget works part time in a shoe store. Sometimes when it is not busy, she rearranges the shoes for fun. If she takes six different pairs of shoes and rearranges them in a row, in how many ways can she rearrange them so that no two shoes match?

Question

Is the answer 478 955 520 or 479 000 160?

Answer 1

Number the pairs 11 22 33 44 55 66

Now, the left shoes can be arranged in 6! ways.

For each of those ways, there are 6! ways to arrange the right shoes. Of those, there are

5! arrangements with one matching pair
4! with two matches,
...
1! with 6 matches.

So, it seems to me like there are

6!(6!-(5!+4!+3!+2!+1!)) = 408 240 ways to do the job.

This is so much smaller than your suggestions, I'd like to see your reasoning.