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

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
\(n=7\)