The number of ways you can choose c items out of a population of n, written:
shows up in many places. For example, one of the authors lives in a state whose lottery involves guessing which 6 numbers between 1 and 59 the lottery authority will pick in the next drawing—that is, the chances of winning depend on how many ways there are to pick 6 numbers from a population of 59. There are many ways to evaluate Equation 15.42, one being to use the following recurrence relation:
Just as with Fibonacci numbers, this recurrence can be directly transcribed into a algorithm, and just as with Fibonacci numbers that algorithm has an exponential execution time. Use dynamic programming to design an algorithm that is based on Equation 15.43, but that runs in less than exponential time. Hint: consider using a two-dimensional array to save values of
for m ranging from 0 to n and k ranging from 0 to c.
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