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# Binomial Coefficients

Binomial coefficients in discrete mathematics are denoted $${n \choose k}$$, where we say $$n$$ choose $$k.$$ The variable, $$n,$$ is usually known as the upper index while the variable, $$k,$$ is known as the lower index. Binomial coefficients are the same amount of combinations of $$k$$ items that can be chosen from a set of $$n$$ items where order does not matter. Therefore, $${n \choose k}$$ gives the $$k$$ subsets that are possible out of $$n$$ total items. When looking further into binomial coefficients, we notice that the values are non-negative.
The reason why these numbers are called binomial coefficients is because the numbers themselves appear as coefficients in the expansion of a binomial, $$\left(1+d\right)^n,$$ in growing powers of $$d.$$
There are also a few good things to take note of when looking at an expansion:
• There are $$n+1$$ terms in the expansion $$\left(x+y\right)^n.$$
• The degree of each term is $$n.$$
• The powers on $$x$$ begin with $$n$$ and decrease to $$0.$$
• The powers on $$y$$ begin with $$0$$ and increase to $$n.$$
• The coefficients are symmetric.
In addition, there are also numerous identities that are satisfied by binomial coefficients such as: