Task B-4.4. The sum of all coefficients in the expression $(1+x)^{n}+(1+x)^{n+1}$ is equal to 1536. Determine the coefficient of $x^{6}$.
Solution. We introduce the notation $P(x)=(1+x)^{n}+(1+x)^{n+1}$. The sum of the coefficients in the expansion of this polynomial is equal to the value of the polynomial at 1.

(1 point)

$$
\begin{aligned}
P(1) & =(1+1)^{n}+(1+1)^{n+1}=2^{n}+2^{n+1}=3 \cdot 2^{n} \\
3 \cdot 2^{n} & =1536 \\
2^{n} & =512 \\
n & =9
\end{aligned}
$$

In the expansion of $(1+x)^{9}$, the coefficient of $x^{6}$ is $\binom{9}{6}$, and in the expansion of $(1+x)^{10}$, the coefficient of $x^{6}$ is $\binom{10}{6}$. (1 point)

Then the coefficient of $x^{6}$ in the polynomial $P(x)$ is $\binom{9}{6}+\binom{10}{6}=294$.