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: