N! gives the number of permutations of ‘n’ objects from a set of ‘n’ distinct objects. = nP r (notation for the number of permutations of r objects out of a set of n distinct objects) A Specific Case Thus, from the Product Rule of Counting, we can get, Similarly, for the third step, we have ‘(n-2)’ objects available to us. In the next event, however, we have ‘(n-1)’ objects available for choice, since we must not include the object that we have already chosen in the first step. For example, when we begin the selection, for the first object, we have all the ‘n’ choices available to us. Hence our choices after each event get reduced by one. In this case, when the repetition of objects is not allowed, we must be careful, not to choose a specific object more than once. This is the permutation formula for calculating the number of permutations possible for the choice of ‘r’ items from a set of ‘n’ distinct items when repetition is allowed. Using the fundamental principle then, we get, Since this event is taking place ‘r’ times and the act of choosing an item from the available set is always independent of our other choices, we may invoke the Product Rule of Counting here. N(E) = n (the number of ways in which E can take place) Let us call the event of choosing an item as E: So, for choosing ‘r’ items, we have n choices available to us ‘r’ times. When the repetition of items is allowed, at every step of selection from the set of ‘n’ items, we have all the ‘n’ choices available to us since we can make a choice multiple times. Selection when the repetition of items is not allowed.Selection when the repetition of items is allowed.We can choose the required items in the following two ways: Let’s say we have a set of ‘n’ distinct items, out of which we must choose ‘r’ items. (Image Source: Wikipedia) Types of Permutation Since we have already studied combinations, we can also interpret permutations as ‘ordered combinations’. In other words, a permutation is an arrangement of objects in a definite order. A permutation is a collection or a combination of objects from a set where the order or the arrangement of the chosen objects does matter.
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