$\left({ }^nC_0+{ }^n C_1\right)({ }^n C_1+{ }^n C_2)({{ }^nC_2+{ }^nC_3}) \cdots\left({ }^nC_{n-1}+{ }^n C_n\right)$

$=\frac{C_0 C_1 C_2 \ldots C_n}{n !}\times(n+1)C^n$

$={ }^{n+1} C_1{ }^{n+1} C_2{ }^{n+1} C_3 \cdots {}^{n+1} C_n$

$={ }^{n+1} C_1{ }^{n+1} C_2{ }^{n+1} C_3 \cdots {}^{n+1} C_n$

$=\frac{(n+1) n !}{n1 !(n-1) !} \cdot \frac{(n+1) n !}{(n-1) \cdot 2 !(n-2) !} \cdots \cdot \frac{(n+1) n !}{1 \cdot n ! 0 !}=(n+1)^n C_1 C_2 \ldots C_n$

$\begin{aligned} & \frac{C_1}{C_0}+\frac{2 C_2}{C_1}+\frac{3 C_3}{C_2}+\frac{4 C_4}{C_3}+\cdots+\frac{n C_n}{C_{n-1}}=\frac{n(n+1)}{2} \\ & r \cdot \frac{C_r}{C_{r-1}}=\frac{r \cdot n !}{r !(n-r) !} \cdot \frac{(r-1) !(n-r+1) !}{n !}=n-r+1 \\ & n+(n-1)+(n-2)+(n-3)+\cdots+1=\frac{n(n+1)}{2} \end{aligned}$

$C_0+C_{1 / 2}+C_{2 / 3}+C_{3 / 4}+\cdots+C_{n / n+1}=\frac{2^{n+1}-1}{n+1}$

Solution:
$C_1+2 C_2+3 C_3+\cdots n C_n= n \cdot 2^{n-1} \\ n \cdot(1+x)^{n-1}$

$\quad x=1 \\ \left.\int C\right) d x=\int(\quad) d x \leftarrow \text { Not correct } \\ + k_1 \quad \quad \quad + k_2$

$\begin{aligned} & (1+x)^n=C_0+C_1 x+C_2 x^2+C_3 x^3+\cdots+C_n x^n \\ & \int_0^1(1+x)^n d x=\int_0^1\left(C_0+C_1 x+\cdots+C_n x^n\right) d x \\ & {\left[\frac{(1+x)^{n+1}}{n+1}\right]_0^{1}=\frac{2^{n+1}}{n+1}-\frac{1}{n+1}} \end{aligned}$
$\begin{aligned} & =\int_0^1 C_0 d x+\int_0^1 C_1 x d x+\int_0^1 C_2 x^2 d x+\cdots+\int_0^1 C_n x^n d x \\ & =C_0+\frac{C_1}{2}+\frac{C_2}{3}+\cdots+\frac{C_n}{n+1}\end{aligned}$

$2 C_0+2^2 \frac{C_1}{2}+2^3 \frac{C_2}{3}+2^4 \frac{C_3}{4}+\cdots+2^{n+1} \frac{C_n}{n+1} =\frac{3^{n+1}-1}{n+1}$

Solution:

$\int_0^2 C_0 dx+\int_0^2 C_1 xdx+\int_0^2 C_2 x^{2 }dx+\ldots \int_0^2 C_n x^n d x \quad =\int_0^2(1+x)^n d x = \left[\frac{(1+x)^{n+1}}{n+1}\right]^2_0$

$=2 C_0+\frac{2^2 C_1}{2}+\frac{2^3 C_2}{3}+\cdots+\frac{2^{n+1} C_n}{n+1}$

$\begin{aligned} & C_0-\frac{C_1}{2}+\frac{C_2}{3}-\cdots+(-1)^n \frac{C_n}{n+1}=\frac{1}{n+1} \\ & \int_0^1(x-1)^n d x=\left[\frac{(x-1)^{n+1}}{n+1}\right]_0^1\end{aligned}$

Find the coefficient of $x^{49}$ in $\left(x-C_1 / C_0\right)\left(x-2^2 C_{2 / C_1}\right)\left(x-3^2 C_3 / C_2\right) \cdots\left(x-50 C_{49}^2)\right.$

Solution:

$= -\frac{C_1}{C_0}-2^2 \frac{C_2}{C_1}-3^2 \frac{C_3}{C_2} \ldots .-50^2 \frac{C_{50}}{C_{49}}$

$\Rightarrow r$ th term $=r^2 \frac{C_r}{C_{r-1}}= r^2 \frac{50!}{r!(50 - r)}= (50 - r + 1)$

$\begin{aligned} & 1.50+2 \cdot 49+3 \cdot 48+4 \cdot 47+\cdots{50} \cdot 1 \\ & \sum_{r=1}^{50} r(50-r+1)=\sum_{r=1}^{50} 51 r-\sum_{r=1}^{50} r^2\end{aligned}$

$\begin{aligned} & =51 \sum_{r=1}^{50} r-\sum_{r=1}^{50} r^2 \\ & =51 \times \frac{50 \times 51}{2}-\frac{50 \times 51 \times 101}{6}=22100\end{aligned}$

-22100

Find the cofficient of $t^{24}$ in

$\left(1+t^2\right)^{12}\left(1+t^{12}\right)\left(1+t^{24}\right)$

Solution:

$1+1+{ }^{12} C_6$

${ }^{30}C_0 { }^{30}C_{10}-{ }^{30}C_1{ }^{30}C_{11} +{ }^{30}C_2{ }^{30}C_{12}-{ }^{30}C_3{ }^{30}C_{13}$

$+\ldots+{ }^{30}C_{20} { }^{30}C_{30}$

$C_0 C_r+C_1 C_{r+1}+\ldots+C_{n-r} C_n={ }^{2 n} C_{n+r}$

$\begin{aligned} & (x-y)^{20} (x+y)^{30} (y+x) \longrightarrow \text { Coefficient } of y^{20} x^{40} \text { ? } \\ & { }^{30} C_0 x^{30}\quad{ }^{30} C_{00} y^{20} x^{10} \\ & -{ }^{30} C_1 x^{29} y \quad{ }^{30} C_11 y^{19} x^{11} \\ & +{ }^{30} C_2 x^{28} y^2\quad{ }^{30} C_{12} y^{18} x^{12}={ }^{30} C_{20}={ }^{30} C_{10} \\ & (x-y)^{30}(y+x)^{30}=\left(x^2-y^2\right)^{30} \\ & \end{aligned}$

Thank you