I tried so hard for the question below and I couldn't figure it out. Please show some working out. Thanks
10 owners of pedigree dogs will enter the arena to parade their dogs by walking around a circular track. 3 dogs cannot get along together but their is no problem if any 2 of the 3 are together. The number of way avoiding this problem is?
To solve this problem, we can use the concept of combinations. Let's break it down step by step.
Step 1: Calculate the total number of ways to arrange the 10 owners and their dogs around a circular track.
Since the 10 owners and their dogs will be walking around a circular track, the arrangement will be considered circular permutations. The formula to calculate the number of circular permutations of n objects is (n-1)!
In this case, there are 10 owners and dogs, so the number of circular permutations will be (10-1)! = 9!.
Step 2: Subtract the number of arrangements where the 3 problematic dogs are together.
To calculate the number of arrangements where the 3 problematic dogs are together, we can treat them as a single unit. So, there will be (9-1)! arrangements of these 3 dogs with the remaining 7 owners and their dogs.
However, within this group of 3 dogs, they can be arranged themselves in 3! ways. So, we need to multiply the number of arrangements with the remaining 7 owners and their dogs by 3!.
Therefore, the number of arrangements with the 3 problematic dogs together is (9-1)! * 3!
Step 3: Calculate the number of ways to avoid the problem.
To avoid the problem, we need to subtract the number of arrangements with the 3 problematic dogs together from the total number of arrangements in step 1.
Number of ways to avoid the problem = Total number of arrangements - Number of arrangements with the 3 problematic dogs together
= 9! - (9-1)! * 3!
Now, let's calculate the result.
9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 362,880
(9-1)! * 3! = 8! * 3! = 40,320 * 6 = 241,920
Number of ways to avoid the problem = 362,880 - 241,920 = 120,960
Therefore, there are 120,960 ways to avoid the problem of the 3 dogs not getting along together while walking around the circular track.
Consider the group of 3 dogs as a unit. That means there are 7 friendly dogs, and another "fighting unit" which can be anywhere in the line.
So, there are 8! ways to arrange the dogs which will cause a problem, because the three fighters are all together.
Any other arrangement other than those 8! will be ok. So, there are
10! - 8! ways to line up the dogs so that the three fighters are not together.