Dynamic Programming question: Binomial Coefficient

timer Asked: Dec 7th, 2014

Question description

 A Binomial Coefficient C(n, k) is the number of distinct ways of selecting k objects from a set of n objects. There is only 1 way to select all n objects and only 1 way to select 0 objects. If we select the n’th object, then we must select k − 1 objects from the remaining n − 1 objects which can be done in C(n − 1, k − 1) ways. If we don’t select the n’th object, then we must select all k objects from the remaining n − 1 objects which can be done in C(n − 1, k) ways. These observations lead to the following recursive formula.

C(n, k) = {  

1 if k = 0, or k = n 

C(n − 1, k − 1) + C(n − 1, k) otherwise

Using this recursive formula in a straightforward recursive program would be horribly inefficient because you would be solving the same subproblems over and over again. Instead use this recursive formula to come up with a dynamic programming solution. What is the complexity of your algorithm in terms on n and k?

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