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If $n$ is even and the value of ${ }^n C_r$ is maximum, then $r=$
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Verified Answer
The correct answer is:
$\frac{n}{2}$
Apply the formula for binomial coefficient series,
${ }^n C_0,{ }^n C_1,{ }^n C_2, \ldots,{ }^n C_r, \ldots$, ${ }^n C_{n-2},{ }^n C_{n-1},{ }^n C_n$
Here, we have to find the maximum value of value of ${ }^n C_r$
And maximum value will be the middle binomial coefficient.
Total number of terms $=n+1 ; r=0$ to $n$
If we compare, ${ }^n C_r$ is $(r+1)$ th term And middle term
$=\frac{(n+1+1)}{2}=\frac{n}{2}+1$
$\Rightarrow r+1=\frac{n}{2}+1$
$\Rightarrow \quad r=\frac{n}{2}$
${ }^n C_0,{ }^n C_1,{ }^n C_2, \ldots,{ }^n C_r, \ldots$, ${ }^n C_{n-2},{ }^n C_{n-1},{ }^n C_n$
Here, we have to find the maximum value of value of ${ }^n C_r$
And maximum value will be the middle binomial coefficient.
Total number of terms $=n+1 ; r=0$ to $n$
If we compare, ${ }^n C_r$ is $(r+1)$ th term And middle term
$=\frac{(n+1+1)}{2}=\frac{n}{2}+1$
$\Rightarrow r+1=\frac{n}{2}+1$
$\Rightarrow \quad r=\frac{n}{2}$
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