Question

The value of $3^7{C}_0+4^7{C}_1+5^7{C}_2+-----------{10}^7{C}_7$ is ------------

• A. $10(2{)}^6$
• B. $13(2{)}^7$
• C. $14(2{)}^6$
• D. $13(2{)}^6$

Answer 1

The correct option is A $10(2{)}^6$

Given that,

$3^7{C}_0+4^7{C}_1+5^7{C}_2+..............{.10}^7{C}_7$

$t_n={\left(r+3\right)}^n{C}_r$

We know that,

$\left[r^n{C}_r=n{.}^{n-1}{C}_r\right]$

$S_n=\sum _{r=0}^7+n=\sum _{r=0}^7{r}^n{C}_r+3\sum _{r=0}^7{}^n{C}_r$

$=\sum _{r=0}^7{7}^6{C}_r+3\sum _{r=0}^7{}^7{C}_r$

$=7\times 2^6+3\times 2^7$

$=13{\left(2\right)}^6$

Then,

Option $A$ is correct answer.