3.0 Properties of Binomial Expansion

1. If ${}^n{C_x} = {}^n{C_y}$, then either $x=y$ or $x+y=n$. $$ \Rightarrow {}^n{C_r} = {}^n{C_{n - r}} = \frac{{n!}}{{r!\left( {n - r} \right)!}}$$

2. The binomial coefficient in the expansion of ${(a + x)^n}$ which are equidistant from the beginning and the end are equal i.e., $$^n{C_r}{ = ^n}{C_{n - r}}$$

From the above expression, we can write $$\begin{equation} \begin{aligned} ^n{C_0}{ = ^n}{C_n} = 1 \\ ^n{C_1}{ = ^n}{C_{n - 1}} = n \\ ^n{C_2}{ = ^n}{C_{n - 2}} = \frac{{n(n - 1)}}{{2!}}{\text{ and so on}}{\text{.}} \\\end{aligned} \end{equation} $$

3. In each term the sum of indices of $x$ and $y$ is equal to $n$.