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?

Is the answer 478 955 520 or 479 000 160?

To solve this problem, we can use the principle of permutations, specifically, the permutation of objects with repetition.

Let's consider arranging the shoes in a row, one by one.

For the first shoe, Bridget has six options to choose from.
For the second shoe, since it cannot be the same as the first shoe, she has five options left.
For the third shoe, she has four options left, and so on.

Using the multiplication principle, the total number of ways Bridget can rearrange the shoes so that no two shoes match is given by:

6 x 5 x 4 x 3 x 2 x 1 = 720.

Therefore, the answer is 720.

Neither 478,955,520 nor 479,000,160 is the correct answer.

To find the number of ways Bridget can rearrange the shoes so that no two shoes match, we can use the concept of permutations.

The total number of ways Bridget can arrange the shoes without any restrictions is given by the factorial of the total number of shoes. In this case, there are 6 pairs of shoes, so there are 12 individual shoes in total. Therefore, the total number of arrangements is 12!

However, since we don't want any two shoes to match, we need to account for the cases where two shoes of the same pair end up next to each other.

To count these cases, we can treat each pair of shoes as a single entity or unit. This means that instead of having 12 individual shoes, we now have 6 pairs. Since we have 6 units, the total number of arrangements is 6!.

Therefore, the number of ways Bridget can rearrange the shoes so that no two shoes match is:
12! - 6!

Now, let's calculate the answer:

12! = 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 479,001,600
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

So, the number of ways Bridget can rearrange the shoes is:
479,001,600 - 720 = 478,955,880

Therefore, the correct answer is 478,955,880, not 478,955,520 or 479,000,160.

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.