Which ones are combinations and which ones are permutations....and tell why!?

Question

I have 4 questions and I wanted to know if they they were a permutation or combination.

1) how many different breakfasts consisting of juices, cereal, and eggs are available in a cafeteria that served 5 kinds of juices, 6 kinds of cereal, and 2 kinds of eggs?

2) how many ways can a 5-question true-false quiz can be answered?

3) the manager of an accounting department wants to form a three-person advisory committee from the 16 employees in the department. In how many ways can the manager do this?

4) a states department of transportation plans to develop a new section of interstate highway and receives 16 bids for the project. The state plans to hire four of the bidding companies. How many different combinations of four companies can be selected from the 16 bidding companies.
25 minutes ago - 4 days left to answer

Answer 1

A COMBINATION is a collection of objects in which order doesn't matter whereas a PERMUTATION is a collection of objects in which order does matter.



1. Does it ask you to drink the juice, the cereal and the eggs in a particular order? No. Therefore, all that matters is what you choose for a juice, a cereal and an egg. COMBINATION.



2. I will admit that this question is misleading in that you could answer question #1, then 4, then 3, then 5 and then 2 or you could 1-2-3-4 5 which would imply a permutation as order is important.



However, as it's a True False quiz, what the question is most likely asking is the number of possible True-False arrangements for answering questions 1 to 5 in order. Key word is order. Therefore, this is a PERMUTATION as T-F-T-T-T is counted as different from T-T-T-F-T.



3. If you have Bob, Jen and Ray on the same advisory committee and it doesn't matter who was picked first, second or third, then order isn't important. COMBINATION



4. Does it matter in which the bids are selected or just which four are selected? I will leave this one to you.

Answer 2

None of these questions is a permutation question, because the order in which choices are made does not affect the outcome. The first two questions represent a completely different kind of combination question than the last two do, though. They are independent-choice problems, in which the choices do not depend on the results of other choices. For instance, what kind of cereal you can choose does not depend on what kind of juice you have chosen, etc. Independent-choice problems are solved by multiplying together the various numbers of possibilities. To answer problem 2, for example, you multiply 2 * 2 * 2 * 2 * 2, since there are five independent choices with two possibilities for each.



The last two problems are classical combination problems. For example, if the manager chooses one person, the next choice is limited to the remaining people. On the other hand, they are not permutation problems, because the order in which choices were made does not matter to the outcome. If companies A, B, C, and D are chosen in problem 4, it doesn't matter if they were chosen in that order, or in the order B, D, C, A, or in some other order. Combination problems like 3 and 4 are solved using C(n, k) (or nCk, or however you prefer to write it). The solution to problem 3, for example, is C(16, 3) = (16 * 15 * 14)/(1 * 2 * 3) = 560.



I hope that tells you what you need to know.

Answer 3

1) This is neither Permutation nor Combination. It is simple multiplication of the number of (unrelated) options available for each "slot" in a possible meal. 5 juices, times 6 cereals, times 2 eggs, equals 60 kinds of breakfasts.



2) This is also neither Permutation nor Combination. It is 2 (the number of choices in a true-false question), raised to the power of 5 (the number of questions), which gives 32 possibilities.



3) This is a Combination of 16 things, 3 at a time (because nothing in the statement of the problem says anything about the order of the 3 people mattering). This is 16!/(3!(16-3)!) = (16×15×14)/(3×2×1) = 560 ways



4) This is a Combination of 16 things, 4 at a time (because, again, nothing is said about order mattering). This is 16!/(4!(16-4)!) = (16×15×14×13)/(4×3×2×1) = 1820 ways

Answer 4

It all comes down to order. Recall, a combination is the number of ways to group independent objects while a permutation is the number of ways to order those objects subject to certain conditions.



As stated, (1) and (2) are neither but rather just simple counting questions using the multiplication principle. For (3), does it matter in which ORDER the people are selected for this committee or is it just asking the number of ways to find GROUPS of 3 people out of 16? For (4), ask yourself the same question that I stated for (3).