The binomial coefficient series is

${ }^n C_0,{ }^n C_1,{ }^n C_2, \ldots,{ }^n C r, \ldots .,{ }^n C_{n-2},{ }^n C_{n-1},{ }^n C_n$

the task is to find maximum value of ${ }^{ n } Cr$

The value of coefficients increasing initially and reaches its maximum and then start reducing value in middle is maximum

total term are $=n+1$

${ }^n Cr$ is $r +1$ th term

middle term $=(n+1+1) / 2=n / 2+1$

$r+1=n / 2+1$

$r=n / 2$

If $n$ is even and the value of ${ }^n C r$, is maximum, when $r=n / 2$

if $n$ is odd, the maximum occurs at $r=(n-1) / 2$ and at $r=(n+1) / 2$.

$r=n / 2$ if $n$ is Even