#Permutations vs combinations series#

Therefore, the answer is 4 x 3 x 2 x 1 = 24. Question: How many ways can we arrange 4 people A, B, C, D to sit in a different order?Īnswer: For the first seat, we have 4 choices, for the second seat, we have 3 choices, for the third seat, we have 2 choices, and for the last seat, 1 choice. Next, we’ll look at a different kind of question: Explaining Permutation to Young Students Another common model is the tree of possibilities, branching at each decision to be made.įor more examples, see: Examples of the Fundamental Counting Principle Counting Answer Keys Permutations of an entire set In this kind of problem, there is a fixed number of choices at each step, so we multiply those numbers. The number of boxes is 3 rows times 2 columns = 6 boxes.

Mary can put a check in any of the 6 boxes in the table, and that's her choice - the row shows which ribbon she picked, and the column shows which paper she picked. But why do you multiply? We can write the choices as a table, like this: This shows up elsewhere in math, from probability to logic. Note that if we take the question as an “or” (paper or ribbon), we would add but it is intended as an “and” (paper and ribbon), and that makes us multiply. She has probably wrapped a lot of presents, so this was easy for her. If Mary does it this way, she has these choices: (Once my brother built a greenhouse for my father, and he wrapped it with a huge ribbon but no wrapping paper.) Normally, though, we wrap it in paper AND a ribbon. But that's not how we usually wrap gifts. If she picks ONE of these 5 things to wrap the gift, she has 5 ways to do it. How many many different ways can she wrap the gift?ĭoctor Rick answered, first just showing what answer is correct by actually doing the counting, and in passing showing how a wrong interpretation of the question could result in the wrong answer: Let's try it. She has 3 different colors of ribbon and 2 different colors of wrapping paper. I think I should add but my mom said to multiply. Here is a question about that, from 1999: Gift Wrap Combinations

#Permutations vs combinations series#

We can begin with what is sometimes called the Fundamental Principle of Counting: When we have a series of choices, with m ways to make the first choice, n ways to make the second choice, and so on, the total number of ways to make all the choices is the product of those numbers.

Then we’ll go beyond mere formulas, looking into some challenging problems and helpful ways of thinking. I want to start with some questions about the basics, developing the concepts of permutation and combination, and seeing where the formulas for them come from. I want to explore various aspects of this field over the next week or two. As a result, there is no one category to put it into without turning away suitable readers. These topics are studied at all levels of mathematical education, from elementary (where they might just be called counting) to high school (where they are often learned along with probability) to college (where they are part of “discrete math”). We have seen a number of questions recently about combinatorics: the study of methods for counting possibilities.