Isye 2027

# This is equal to nkk and is called n choose k and is

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Unformatted text preview: out labels. Thus, the number of binary sequences 9! of length 9 having ﬁve ones is 5!4! . This is the same as the solution to Example 1.3.5. In fact, there is a one-to-one correspondence between lineups and binary sequences of length 9 with 5 ones. The positions of the 1’s indicate which players are in the lineup. The sequence 110110010 corresponds to the lineup {A, B, D, E, H }. In general, the number of subsets of size k of a set of n distinct objects can be determined as follows. There are n ways to select the ﬁrst object, n − 1 ways to select the second object, and so on, until there are n − k + 1 ways to select the k th object. By the principle of counting, that gives a total count of n(n − 1) · · · (n − k + 1), but this chooses k distinct objects in a particular order. By deﬁnition of the word “set,” the order of the elements within a set does not matter. Each set of k objects is counted k ! ways by this method, so by the principle of over counting, the number of subsets of...
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