![]() ![]() (You will see this as a Binomial distribution in future, but it follows directly from combinatorics as you can see above. The fundamental difference between permutation and combination is the order of objects, in permutation the order of objects is very important, i.e. So, our net size of the set that we care about is: How many ways can we choose exactly $k$ balls out of $N$? That is $\binom$. Finding the distinction between permutation & combination can be done with the aid of the permutation and. Additionally, we use permutations to determine the amount of potential combinations of unrelated objects. Now, how many of these outcomes do we care about? We only care about those that have exactly $k$ $A$s in them. Combinations are used to group objects or to determine how many subgroups can be formed from the given collection of objects. We are going to "normalize" all the sets by this factor, so that the set that contains all the outcomes has size 1. The key difference between these two concepts is ordering. Im going to introduce you to these two concepts side-by-side, so you can see how useful they are. ![]() (For compactness, we can represent this sentence as (A,A,C.,A).) Now, how many such outcomes are there? For each ball, there are three choices, and there are N balls, so there are $3^N$ outcomes. Permutations and Combinations are super useful in so many applications from Computer Programming to Probability Theory to Genetics. To do this question, we should think about how many outcomes we could have had: We can label each outcome as "First ball went to A, second ball went to A, third ball went to C. Now, about this experiment, we can ask: "What is the probability that there are k balls in bucket A?" Let's say we are tossing N balls to three buckets: A, B, and C, and each ball has an equal chance of landing in each bucket. determine the sizes of certain sets.ĮDIT: After the comment by the OP, I decided to add an example: given the question, I expect OP wants an example other than die and coin tosses, so here is one: (When we say that some event has probability a half, we actually mean that the set of outcomes that constitute that event have a "size" of 1/2.) Permutations and combinations allow you to count, i.e. Probability is -fundamentally- about sizes of certain sets. ![]()
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