7.0 Greatest binomial coefficient

To determine the greatest coefficient in the binomial expansion of ${(1 + x)^n}$ where $n$ is a positive integer, Coefficient of $$\left( {\frac{{{T_{r + 1}}}}{{{T_r}}}} \right) = \frac{{{}^n{C_r}}}{{{}^n{C_{r - 1}}}} = \frac{{n - r + 1}}{r} = \frac{{n + 1}}{r} - 1$$

Now, the ${\left( {r + 1} \right)^{th}}$ binomial coefficient will be greater than the ${\left( {r} \right)^{th}}$ binomial coefficient when $$\begin{equation} \begin{aligned} {T_{r + 1}} \geqslant {T_r} \\ \Rightarrow \frac{{n + 1}}{r} - 1 \geqslant 1 \\ \Rightarrow \frac{{n + 1}}{2} \geqslant r\quad ...(1) \\\end{aligned} \end{equation} $$

where $r$ must be an integer.

When $n$ is even, the greatest binomial coefficient is given by the greatest value of $r$ i.e., $r = \frac{n}{2}$ and hence the greatest binomial coefficient is ${}^n{C_{\frac{n}{2}}}$. Similarly, when $n$ is odd, the greatest binomial coefficient is given when $r = \frac{{n - 1}}{2}{\text{ or }}\frac{{n + 1}}{2}$ and hence the greatest binomial coefficient is $^n{C_{\frac{{n + 1}}{2}}}{\text{ or}}{{\text{ }}^n}{C_{\frac{{n - 1}}{2}}}$, both of the values are equal.

The above result concluded that the greatest binomial coefficient is the binomial coefficient of the middle term.