Probability of you getting at least 2 heads is 2 outcomes / 4 Combinations (with Repetition) = 0.5. If you are looking for "at least 2 Heads", 2 options match: HHH and HHT (order not important). These are (because order is not important): HHH, HHT, HTT, TTT If the question is "If you throw a 2-sided coin (N=2), R times, how many times can you get at least 2 heads?", you are looking for Combination (order is not important) with Repetition where "HHT" and "THH" are same outcomes (combination).Ĭombination with Repetition formula is the most complicated (and annoying to remember): (R+N-1)! / R!(N-1)!įor 3 2-sided coin tosses (R=3, N=2), Combination with Repetition: (3+2-1)! / 3!(2-1)! = 24 / 6 = 4 Probability of "at least 2 heads in a row" is 3/8th (0.375) In these, "at-least-2 Heads in a row" permutations are: HHH, HHT, THH - 3. Permutation with Repetition is the simplest of them all:ģ tosses of 2-sided coin is 2 to power of 3 or 8 Permutations possible. If the question is "How many ways a series of R coin tosses (N=2 sides) can go? Of these, how many will have 2 Heads in the row?", you are looking for Permutation with Repetition where "HHT" is different outcome from "THH". *Probably the best page that summarizes the Combination vs Premutation with or without Repetition * We offer tutoring programs for students in K-12, AP classes, and college.Coin toss series can be viewed, depending on what you want to know, as either "combination" or "permutation" but in all cases "with repetition" (meaning same side can occur again and again). SchoolTutoring Academy is the premier educational services company for K-12 and college students. Interested in algebra tutoring services? Learn more about how we are assisting thousands of students each academic year. If all runners were given places, that would lead to 11! places, or 39,916,800 possible combinations. There are 11 possibilities for first place, 10 possibilities for second place, and 9 possibilities for third place, or 990 permutations in the first 3 places. Most of the time, all 11 runners aren’t given places, just the first 3. How many of the groups of 5 contain the same members in whatever order? That can be answered by the factorial 5!, which is 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 120. Since the order does matter, it will be a subset of the 30240. Unlike permutations, order does not count. So the only difference between the two formulas is that nCr has an additional r in the denominator (that is the number of ways in which you can arrange r. If the order mattered, it would be 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 = 30240. In other words, how many different combinations of two pieces could you end up with In counting combinations, choosing red and then yellow is the same as choosing yellow and then red because in both cases you end up with one red piece and one yellow piece. In this case, it doesn’t matter in what order they are chosen, because the end result is the same. For example, suppose 5 people are chosen out of a group of 10 to be on a committee. It is a mathematical convention that makes calculations easier.Ĭombinations are rankings where the order doesn’t matter. The number 0! is equal to 1, much in the same way that x 0 is equal to 1. In the example where there were 10 songs on a side of the CD, 10! (read 10 factorial) is 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1. They are multiplications that start with the number of events multiplied by the number of events -1 and so on, until there is only 1 left. The way to calculate the number of permutations is 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 3, 628, 800.įactorials involve all possible outcomes of an event. For the third track, there are 8 left, and so on, until the last song remains. Since one song has already been chosen, there are 9 left for the second track. If there are 10 songs that have been recorded, there are 10 possibilities for the opening track. Suppose a music producer is choosing the order of songs on a CD released from a new artist. Combination is defined and given by the following function. we might ask how many ways we can select 2 letters from that set. For example, suppose we have a set of three letters: A, B, and C. Permutations are rankings where the order matters. A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected. If items are ordered in a particular way, factorials determine the number of times they can be ordered. Permutations relate to the order of objects, while factorials involve all possible outcomes of an event. Permutations and factorials are closely related mathematical concepts.
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