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As can be seen, the first choice was for A to be captain out of the 11 initial members, but since A cannot be the team captain as well as the goal keeper, A was removed from the set before the second choice of the goal keeper B could be made. Thus, the generalized equation for a permutation can be written as:. Again, the calculator provided does not calculate permutations with replacement, but for the curious, the equation is provided below:. Combinations are related to permutations in that they are essentially permutations where all the redundancies are removed as will be described below , since order in a combination is not important.
Combinations, like permutations, are denoted in various ways including n C r , n C r , C n,r , or C n,r , or most commonly as simply n r. As with permutations, the calculator provided only considers the case of combinations without replacement, and the case of combinations with replacement will not be discussed.
Using the example of a soccer team again, find the number of ways to choose 2 strikers from a team of Unlike the case given in the permutation example, where the captain was chosen first, then the goal keeper, the order in which the strikers are chosen does not matter, since they will both be strikers.
Referring again to the soccer team as the letters A through K , it does not matter whether A and then B or B and then A are chosen to be strikers in those respective orders, only that they are chosen. The possible number of arrangements for all n people, is simply n! To determine the number of combinations, it is necessary to remove the redundancies from the total number of permutations from the previous example in the permutations section by dividing the redundancies, which in this case is 2!.
Again, this is because order no longer matters, so the permutation equation needs to be reduced by the number of ways the players can be chosen, A then B or B then A , 2, or 2!. This yields the generalized equation for a combination as that for a permutation divided by the number of redundancies, and is typically known as the binomial coefficient:. For this problem we are looking for an ordered subset of 3 horses r from the set of 4 best horses n.
We are ignoring the other 11 horses in this race of 15 because they do not apply to our problem. We must calculate P 4,3 in order to find the total number of possible outcomes for the top 3 winners. At a high school track meet the meter race has 12 contestants. The top 3 will receive points for their team. How many different permutations are there for the top 3 from the 12 contestants? For this problem we are looking for an ordered subset 3 contestants r from the 12 contestants n.
We must calculate P 12,3 in order to find the total number of possible outcomes for the top 3. An NFL team has the 6th pick in the draft, meaning there are 5 other teams drafting before them. If the team believes that there are only 10 players that have a chance of being chosen in the top 5, how many different orders could the top 5 be chosen?
For this problem we are finding an ordered subset of 5 players r from the set of 10 players n.