feb6 - Illinois Institute of Technology Department of...

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Illinois Institute of Technology Department of Computer Science Lecture 5: February 6, 2008 CS 330 Discrete Structures Spring Semester, 2008 1 How big is n ! ? In the preceding material we have seen that n ! occurs in many contexts when counting arrangements of elements. Furthermore, in subsequent sections we will ±nd that it ±gures centrally in most of the counting problems that we pursue. It is reasonable, then, to ask about the behavior of n ! as a function n . Speci±cally, how quickly does n ! grow as n becomes large? The answer to this question will give us asymptotic information about how the number of con±gurations grows in various cases; we will use such information in the next section, for example, to establish a benchmark for the performance of sorting algorithms. Brief computation reveals that n ! grows fast as n increases: n 0 1 2 3 4 5 6 7 8 9 10 n ! 1 1 2 6 24 120 720 5,040 40,320 362,880 3,628,800 For example, if a code-cracking program requires the examination of each of the permutations of n items, the program will (most likely) be practical for
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feb6 - Illinois Institute of Technology Department of...

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