G Tewani Solutions for Chapter: Binomial Theorem, Exercise 4: DPP

If the value of ${}^{\mathrm{n}}\mathrm{C}_0+2{·}^{\mathrm{n}}{\mathrm{C}}_1+3{·}^{\mathrm{n}}{\mathrm{C}}_2+\dots +(\mathrm{n}+1){·}^{\mathrm{n}}\mathrm{Cn}=576$, then $\mathrm{n}$ is

The value of $\left(\begin{array}{c}50\\ 6\end{array}\right)-\left(\begin{array}{c}5\\ 1\end{array}\right)\left(\begin{array}{c}40\\ 6\end{array}\right)+\left(\begin{array}{c}5\\ 2\end{array}\right)\left(\begin{array}{c}30\\ 6\end{array}\right)-\left(\begin{array}{c}5\\ 3\end{array}\right)\left(\begin{array}{c}20\\ 6\end{array}\right)+\left(\begin{array}{c}5\\ 4\end{array}\right)\left(\begin{array}{c}10\\ 6\end{array}\right)$ where $\left(\begin{array}{l}\mathrm{n}\\ \mathrm{r}\end{array}\right)$ denotes ${}^{\mathrm{n}}\mathrm{C}_{\mathrm{r}}$ is

The value of the expression $\underset{0\le i<j\le n}{\sum \sum }{\left(-1\right)}^{i+j-1}{·}^n{C}_i{·}^n{C}_j=$

$\sum _{\mathrm{m}=1}^{\mathrm{n}}\left(\sum _{\mathrm{k}=1}^{\mathrm{m}}\left({\sum _{\mathrm{p}=\mathrm{k}}^{\mathrm{m}}}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{m}}{·}^{\mathrm{m}}{\mathrm{C}}_{\mathrm{p}}{·}^{\mathrm{p}}{\mathrm{C}}_{\mathrm{k}}\right)\right)=$

${}^{n}C_0^{2n}{C}_m{-}^n{C}_1^{2n-2}{C}_m{+}^n{C}_2^{2n-4}{C}_m-\dots =$

Which of the following is/are correct?

The value of $\sum _{k=0}^7\left[\frac{\left(\begin{array}{l}7\\ k\end{array}\right)}{\left(\begin{array}{l}14\\ k\end{array}\right)}\sum _{r=k}^{14}\left(\begin{array}{l}r\\ k\end{array}\right)\left(\begin{array}{l}14\\ r\end{array}\right)\right]$, where $\left(\begin{array}{l}n\\ r\end{array}\right)$ denotes ${}^{n}C_r$, is

The value of ${}^{1000}\mathrm{C}_{50}{+}^{999}{\mathrm{C}}_{49}{+}^{998}{\mathrm{C}}_{48}+\dots {+}^{950}{\mathrm{C}}_0$ is