Suppose X is a binomial distribution with parameters n = 100 and p =1/2 then P(X= r) is maximum when r

Given that in binomial distribution 

$$\begin{gathered} \mathrm{n}=100, \mathrm{p}=\frac{1}{2} \\ \Rightarrow \mathrm{q}=1-\mathrm{p}=1-\frac{1}{2}=\frac{1}{2} \end{gathered}$$

Now $\mathrm{P}(\mathrm{X}=\mathrm{r})={}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{r}} {\mathrm{p}}^{\mathrm{r}} {\mathrm{q}}^{\mathrm{n}-\mathrm{r}}$

                        $$\begin{gathered} ={}^{100}{\mathrm{c}}_{\mathrm{r}} {\left(\frac{1}{2}\right)}^{\mathrm{r}} {\left(\frac{1}{2}\right)}^{100-\mathrm{r}} \\ ={}^{100}{\mathrm{c}}_{\mathrm{r}} {\left(\frac{1}{2}\right)}^{100} \end{gathered}$$

We know that ${}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{r}}$ is maximum at $\mathrm{r}=\frac{\mathrm{n}}{2}$

$\therefore {}^{100}{\mathrm{c}}_{\mathrm{r}}$ is maximum at $\mathrm{r}=\frac{100}{2}=50$

$\therefore \mathrm{P}(\mathrm{X}=\mathrm{r})$ is maximum when r=50