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    \(1\) \(\underline{\text{Binomial Coefficient}}\)

    In mathematics, the binomial coefficient is written as \({n \choose k}\) and can be pronounced as “\(n\) choose \(k\).” Alternatively, binomial coefficients are also sometimes given the notation \(C(n,k)\). In this case, the \(C\) stands for the word “choices” or “combination” (Benjamin, 2009, p. 8). This is because there are \({n \choose k}\) ways of choosing \(k\) elements from a set containing a number of \(n\) elements.

    For example, we can consider the set \(A=\{1,2,3,4\}\). If we wish to know how many subsets of \(2\) can be created using this set, we are essentially asking how many ways there are of choosing \(2\) elements from a set with \(4\) total elements. Therefore, we can identify that \(k=2\) and \(n=4\). Hence, we have \({4 \choose 2}\).

    To calculate such a problem, we typically would want to write out by hand all the possible combinations. Doing so, one would find that there are six pairs of size two subsets, namely \(\{1,2\}\), \(\{1,3\}\), \(\{1,4\}\), \(\{2,3\}\), \(\{2,4\}\), and \(\{3,4\}\). However, it becomes clear to see that when we are dealing with large sets of values, this work can become tedious. Therefore, it is convenient to utilize the following formula:

    \[{n \choose k} = \frac{n!}{(n-k)! \cdot k!}\]

    It’s important to note that when we talk about the binomial coefficient, \({n \choose k}\), we are referring to a subset combination. Therefore, order does not matter. However, there are combinations in which the order does matter and we call such combinations permutations.

    The binomial coefficient \({n \choose k}\) can be used to find the number of possible combinations of things such as:

    • different basketball teams of \(x\) amount of players that can be formed from a group of \(y\) amount of players,

    • the number of ways \(4\) people can be chosen to be on a committee within a club that has \(20\) members,

    • the number of ways you can seat \(10\) people into \(15\) chairs,

    • the number of ways you can choose \(5\) candies from \(8\) multicolored candies,

    • the number of full houses that can be dealt from a standard \(52\)-card deck.

    Pictured (above) are two items incorporated in the list of how to use binomial coefficients.