Series Involving Binomial Coefficients - Practice Questions & MCQ

1. Sum of Binomial Coefficients 

$
\begin{aligned}
& C_0+\mathrm{C}_1+C_2+C_3+\ldots \ldots+C_n=2^n \\
& \text { or } \sum_{r=0}^n{ }^n C_r=2^n
\end{aligned}
$

2. Sum of Binomial coefficients with alternate sign

$
\begin{aligned}
& C_0-\mathrm{C}_1+C_2-C_3+\ldots \ldots+(-1)^n C_n=0 \\
& \text { or } \sum_{r=0}^n(-1)^r{ }^n C_r=0
\end{aligned}
$

3. Sum of the Binomial coefficients of the odd terms \& Sum of the Binomial coefficients of the even terms

Adding the above two series we get,

$
\begin{aligned}
& 2 \cdot\left(\mathrm{C}_0+\mathrm{C}_2+\mathrm{C}_4+\ldots \ldots\right)=2^n \\
& \mathrm{C}_0+\mathrm{C}_2+\mathrm{C}_4+\ldots \ldots=2^{n-1}
\end{aligned}
$
Similarly, on subtracting we get,

$
\mathrm{C}_1+\mathrm{C}_3+\mathrm{C}_5 \ldots \ldots=2^{n-1}
$
Hence, the sum of the binomial coefficients of the odd terms = Sum of the binomial coefficients of the even terms

$
\text { I.e. } C_1+C_3+C_5 \ldots \ldots=C_0+C_2+C_4+\ldots \ldots=2^{n-1}
$