If \( \mathrm{C}_{0}, \mathrm{C}_{1}, \mathrm{C}_{2}, \ldots \mathr...
If \( \mathrm{C}_{0}, \mathrm{C}_{1}, \mathrm{C}_{2}, \ldots \mathrm{C}_{\mathrm{n}} \) are the binomial coefficients in the expansion of \( (1+\mathrm{x})^{\mathrm{n}} \) then prove that :
\( \mathrm{P} \)
\[
\frac{(3 \cdot 2-1)}{2} C_{1}+\frac{3^{2} \cdot 2^{2}-1}{2^{2}} C_{2}+\frac{3^{3} \cdot 2^{3}-1}{2^{3}} C_{3}+\ldots \ldots . .+\frac{3^{n} \cdot 2^{n}-1}{2^{n}} C_{n}=\frac{2^{3 n}-3^{n}}{2^{n}}
\]
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