Question

The value of ${{3}^{7}}{{C}_{0}}+{{4}^{7}}{{C}_{1}}+{{5}^{7}}{{C}_{2}}+..........+{{10}^{7}}{{C}_{7}}$ is__________
A) $10{{\left( 2 \right)}^{6}}$
B) $13{{\left( 2 \right)}^{7}}$
C) $14{{\left( 2 \right)}^{6}}$
D) $13{{\left( 2 \right)}^{6}}$

Answer 1

Hint: Here in this question we have been asked to find the value of the given summation ${{3}^{7}}{{C}_{0}}+{{4}^{7}}{{C}_{1}}+{{5}^{7}}{{C}_{2}}+..........+{{10}^{7}}{{C}_{7}}$ in a simplified form. For answering this question we will use $r{{.}^{n}}{{C}_{r}}={{n}^{n-1}}{{C}_{r}}$ .

Complete step by step answer:
Now considering from the question we have been asked to find the value of the given summation ${{3}^{7}}{{C}_{0}}+{{4}^{7}}{{C}_{1}}+{{5}^{7}}{{C}_{2}}+..........+{{10}^{7}}{{C}_{7}}$ in a simplified form.
For answering this question firstly we will observe the pattern involved.
By observing the pattern we can simply write the ${{r}^{th}}$ term of the given pattern will be ${{t}_{r}}={{\left( r+3 \right)}^{7}}{{C}_{r}}$ .
Hence we can write it simply as ${{3}^{7}}{{C}_{0}}+{{4}^{7}}{{C}_{1}}+{{5}^{7}}{{C}_{2}}+..........+{{10}^{7}}{{C}_{7}}=\sum\limits_{r=0}^{7}{{{\left( r+3 \right)}^{7}}{{C}_{r}}}$ .
By further simplifying the above expression we will have $\Rightarrow \sum\limits_{r=0}^{7}{{{r}^{7}}{{C}_{r}}}+3\sum\limits_{r=0}^{7}{^{7}{{C}_{r}}}$ ,
From the basic concepts of combinations we know that $r{{.}^{n}}{{C}_{r}}={{n}^{n-1}}{{C}_{r}}$ now we will use this to simplify the expression further.
By using the above expression we will have $\Rightarrow 7\sum\limits_{r=0}^{6}{^{6}{{C}_{r}}}+3\sum\limits_{r=0}^{7}{^{7}{{C}_{r}}}$ .
From the basic concepts of combinations we know that $\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}}={{2}^{n}}$ now we will use this formula to simplify the above expression further.
Now by further simplifying the above expression using the formula we will have $\Rightarrow 7\left( {{2}^{6}} \right)+3\left( {{2}^{7}} \right)$ .
Now we will take out common ${{2}^{6}}$ from the expression after doing that we will have $\Rightarrow {{2}^{6}}\left( 7+3\left( 2 \right) \right)$ .
Now by further simplifying this we will have
$\begin{align}
  & \Rightarrow {{2}^{6}}\left( 7+6 \right) \\
 & \Rightarrow 13\left( {{2}^{6}} \right) \\
\end{align}$ .
Therefore we can conclude that the value of ${{3}^{7}}{{C}_{0}}+{{4}^{7}}{{C}_{1}}+{{5}^{7}}{{C}_{2}}+..........+{{10}^{7}}{{C}_{7}}\,$ is $13{{\left( 2 \right)}^{6}}$ .

So, the correct answer is “Option D”.

Note: During the process of answering questions of this type we should be sure with the concepts that we are applying and calculations that we are going to perform in between the steps. Now we can generalize this as $\sum\limits_{r=0}^{n}{{{\left( r+x \right)}^{n}}{{C}_{r}}}={{2}^{n-1}}\left( n+2x \right)$ .