Q75 / 180 · KCET 2013 · Mathematics
If the value of $C_{0}+2 \cdot C_{1}+3 \cdot C_{2}+\ldots+(n+1) \cdot C_{n}=576$, then $n$ is equal to
- Correct answerA7
- B5
- C6
- D9
Solution
Given, $C_{0}+2 C_{1}+3 C_{2}+\ldots+(n+1) C_{n}=576$
We know that,
$\begin{array}{r}
(1+x)^{n}={ }^{n} C_{0}+{ }^{n} C_{1} x+{ }^{n} C_{2} x^{2}+\ldots+{ }^{n} C_{n} x^{n} \\
\Rightarrow x(1+x)^{n}={ }^{n} C_{0} x+{ }^{n} C_{1} x^{2}+{ }^{n} C_{2} x^{3}+\ldots +{ }^{n} C_{n} x^{n+1}
\end{array}$
On differentiating w.r.t. $x$, we get
$\left.={ }^{n} C_{0}+2 \cdot{ }^{n} C_{1} \cdot x+3\right)^{n} C_{2} x^{2}+\ldots+(n+1){ }^{n} C_{n} x^{n}$
On putting $n=1$, we get
$2^{n}+n \cdot 2^{n-1}={ }^{n} C_{0}+2 \cdot{ }^{n} C_{1}+3 \cdot{ }^{n} C_{1}+\ldots +(n+1){ }^{n} C_{n}$ (given)
$\Rightarrow \quad 2^{n-1}(n+2)=2^{6} \times 9=2^{(7-1)} \cdot(7+2)$
On comparing, we get
$n=7$
Hence, the correct answer is option: 'A'.