Blog Post 1

$$1$$ $$\underline{\text{Binomial Coefficient}}$$ 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.