So far I got up to part C and I think I have to maybe divide my answer from part B by some number but am totally confused on how to approach this question.
There are 15 dogs in an obedience class. Five of the dogs are Dalmatians. Aside from that, the rest are all different breeds. Assume that the instructor can only distinguish between the dogs by their breeds.
(In other words, he can’t tell the Dalmatians apart!) At the beginning of class all of the dogs are lined up in a row.a. As far as the instructor can tell, how many ways can the dogs be lined up in a row? (Remember,the Dalmations are indistinguishable.)
b. Assume that the dogs are given sweaters that completely disguise what breed they are. 3 wear yellow sweaters, 4 wear red sweaters, 3 wear blue sweaters, and 5 wear green sweaters. At this point, the instructor can only distinguish between the dogs by their sweater colors. (In other words,
all the dogs with the same color sweater look alike to him!) As far as the instructor can tell, how many ways can the dogs be lined up in a row? (You may assume that the dogs will wear the
sweaters without objection.)c. Assume the sweater scenario in part (b). What is the probability that the instructor sees all of the dogs with the same sweater color sitting next to each other (for example: RRRRYYYGGGGGBBB)?
Answers I have so far:
A. 15 choose 5 * 10 choose 10
B.(15 choose 3)(12 choose 4)(8 choose 3)(5 choose 5)
C. ?
Any hints or help is appreciated thank you.
Best Answer
(a) There are $15!$ total permutations. Because the $5$ Dalmatians are indistinguishable, it doesn't matter what order they are in, the instructor will see the same dogs! Therefore we have to divide our total permutations by $5!$ to account for the number of ways the $5$ Dalmatians can arrange themselves. So, our total number of ways is $\frac{15!}{5!}$
You put ${15 \choose 5} \cdot {10 \choose 10}$ as an answer. This is wrong because this means you are trying to choose 5 objects from a set of 15 indistinguishable objects. As you stated in the question, only the $5$ Dalmatians are indistinguishable, and the rest of the dogs are distinguishable.
(b) This is a similar question as the instructor can now distinguish the dogs based on the color sweater the dog wears. Now we still have $15!$ total permutations. But what's different is that the dogs with similar colored sweaters are indistinguishable. What this means is that the $4$ dogs with red sweaters are indistinguishable, the $3$ dogs with yellow sweaters are indistinguishable, etc. So we have divide again like we did before. So our answer is now $\frac{15!}{3!\cdot 4!\cdot 3!\cdot 5!}$
Again your answer is means that you're assuming all of the dogs to be indistinguishable, when in fact they are not.
(c) Part c is tricky. But a helpful trick to solve hit is to visualize each color group being one entity. So we'd have (Red dogs) | (Yellow dogs) | (Blue Dogs) | (Green Dogs), and each dog would be grouped together with each other dog with the same color sweater as that dog. Now because there are $4$ groups, there are $4!$ total ways to arrange the groups (we treat each group as one entity). So, our probability is $\frac{4!}{15!}$