In how many different arrangements can 5 dogs and 7 cats be placed in a row if the dogs must be next to one another?
consider the 5 dogs as one animal. Then there are 8 places it can be placed with the 7 cats.
Each group of 5 dogs can be arranged in 5! ways, so the final number is
8*5!
Thank you!
To find the number of different arrangements where the dogs must be next to one another, we can consider the group of 5 dogs as a single unit. So, we have 5 + 1 = 6 units (1 group of dogs + 7 cats) to arrange in a row.
Now, we can arrange these 6 units in 6! (6 factorial) different ways. However, within the group of dogs, the 5 dogs can be arranged among themselves in 5! ways as well.
Therefore, the total number of different arrangements is 6! × 5!.
Calculating this, we have:
6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
5! = 5 × 4 × 3 × 2 × 1 = 120
So, the total number of different arrangements is:
720 × 120 = 86,400