G Tewani Solutions for Chapter: Binomial Theorem, Exercise 3: DPP

The coefficient of $x^{50}$ in $\left(x{+}^{101}{C}_0\right)\left(x{+}^{101}{C}_1\right)\dots \left(x{+}^{101}{C}_{50}\right)$ is 

In the expansion of ${\left(1+x\right)}^{70}$ the sum of the coefficients of odd powers of $x$ is

The sum of all the coefficients of the terms in the expansion of $(x+y+z+w{)}^6$ which contains $x$ but not $y$, is

The value of ${}^{12}C_2{+}^{13}{C}_3{+}^{14}{C}_4+\dots {+}^{999}{C}_{989}$ is

If ${\left(1+x+x^2\right)}^{25}=a_0+a_{1}x+a_2{x}^2+...+a_{50}{x}^{50}$, then $a_0+a_2+a_4+...+a_{50}$ is

If the sum of the coefficients in the expansion of ${\left(q+r\right)}^{20}{\left(1+\left(p-2\right)x\right)}^{20}$ is equal to square of the sum of coefficients in the expansion of ${\left[2rqx-\left(r+q\right).y\right]}^{10}$, where $p, q, r$ are positive constants, then:

The sum $S_n=\underset{k=0}{\overset{n}{\Sigma }}{\left(-1\right)}^k{·}^{3n}{C}_k$, where $n=1, 2,....$ is

If $1+\left(1+x\right)+{\left(1+x\right)}^2+\dots +{\left(1+x\right)}^n=\sum _{k=0}^n{a}_k{x}^k$, then which of the following is not true?