Binomial Inside Binomial - Practice Questions & MCQ

Upper Index variable

Let's see some examples to understand how to solve such types of questions

How do we find the summation of the series where the upper index also varies

$
\sum_{\text {e.g., }}^{n=0}{ }^{n+r} C_r={ }^n C_0+{ }^{n+1} C_1+{ }^{n+2} C_2+\cdots+{ }^{n+n} C_n
$

[put value of $r=0,1,2, \ldots \ldots$,upto $n$ ]
By using the property of the binomial coefficient

$
\begin{aligned}
& { }^n C_r={ }^n C_{n-r} \\
& \sum_{\mathrm{r}=0}^{\mathrm{n}}{ }^{\mathrm{n}+\mathrm{r}} \mathrm{C}_{\mathrm{r}}={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}}+{ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{n}}+{ }^{\mathrm{n}+2} \mathrm{C}_{\mathrm{n}}+\cdots+{ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{n}}
\end{aligned}
$
Now as ${ }^n C_n={ }^{n+1} C_{n+1}=1$, so we can write

$
\sum_{\mathrm{r}=0}^{\mathrm{n}}{ }^{\mathrm{n}+\mathrm{r}} \mathrm{C}_{\mathrm{r}}={ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{n}+1}+{ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{n}}+{ }^{\mathrm{n}+2} \mathrm{C}_{\mathrm{n}}+\cdots+{ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{n}}
$
Now first 2 terms can be combined by using the property.

$
\begin{aligned}
& { }^n C_r+{ }^n C_{r-1}={ }^{n+1} C_r \\
& \sum_{\mathrm{r}=0}^{\mathrm{n}}{ }^{\mathrm{n}+\mathrm{r}} \mathrm{C}_{\mathrm{r}}={ }^{\mathrm{n}+2} \mathrm{C}_{\mathrm{n}+1}+{ }^{\mathrm{n}+2} \mathrm{C}_{\mathrm{n}}+\cdots+{ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{n}}
\end{aligned}
$

Now first two terms can again be combined using the same property, and this process can be continued till the last term is also combined $={ }^{2 n+1} C_{n+1}$