Amit M Agarwal Solutions for Chapter: Binomial Theorem, Exercise 5: Target Exercises

$\sum _{0\le i<j\le n}{\sum }^n{C}_i$ is equal to

The value of $\sum _{0\le i}\sum _{<j<k<l\le n}2$ is equal to

The value of $\sum _{1\le i<i}\sum _{\le n-1}{\left(i·j\right)}^n{C}_i·{}^{n}C_j$ is equal to

If $S=\sum _{n=1}^{\infty }\left(\frac{{}^{n}C_0+{}^{n}C_1+{}^{n}C_2+\dots +{}^{n}C_n}{{}^{n}P_n}\right),$ then $S$ is equal to

The value of $\sum _{r=0}^{n}r\left(n-r\right)·{\left({}^{n}C_r\right)}^2$ is equal to

If ${}^{n+1}C_{r+1}:{}^{n}C_r:{}^{n-1}C_{r-1}=11:6:3,$ then $nr$ is equal to

If $f\left(x\right)=x^n,$ then the value of $f\left(1\right)+\frac{f^1\left(1\right)}{1!}+\frac{f^2\left(1\right)}{2!}+\dots +\frac{f^n\left(1\right)}{n!},$ where $f^r\left(x\right)$ denotes the $r^{\text{th}}$ order derivative of $f\left(x\right)$ with respect to $x,$ is

If $\alpha =\frac{5}{2!3}+\frac{5·7}{3!3^2}+\frac{5·7·9}{4!3^3}+\dots ,$ then ${\alpha }^2+4\alpha$ is equal to