Binomial Theore...

If $$x+y= 1$$ then $$\displaystyle\sum_{r= 0}^{n} r^{n}C_{r}x^{r}y^{n-r}$$ equals

If $$A$$ is the sum of the odd terms and $$B$$ the sum of even terms in the expansion of $${ \left( x+a \right)  }^{ n }$$, then $${ A }^{ 2 }-{ B }^{ 2 }=$$

If $$\displaystyle \left ( 1+x+x^{2} \right )^{n}=\sum_{r=0}^{2n} a_{r}x^{r}=a_{0}+a_{1}x+a_{2}x^{2}+...+a^{2n}x^{2n}$$ and
$$ \displaystyle P=a_{0}+a_{3}+a_{6}+...$$
$$ \displaystyle Q=a_{1}+a_{4}+a_{7}+...$$
$$ \displaystyle R=a_{2}+a_{5}+a_{8}+...$$
then the set of values of $$ P, Q, R $$ are respectively equals

The evaluated value of $$\displaystyle \sum_{i=0}^{n} \sum_{j=1}^{n}$$ $$\displaystyle ^{n}C_{j}$$ $$\displaystyle ^{j}C_{i},$$ $$\displaystyle i\leq j$$

If $$C_{0},C_{1},C_{2}....,C_{n}$$ are Binomial Coefficients, such that $$\displaystyle S_{n}=\sum_{r=0}^{n}\frac{1}{{C_{r}}^{n}}$$ and $$\displaystyle t_{n}= \sum_{r= 0}^{n}\frac{r}{{C_{r}}^{n}}$$ then $$\displaystyle \frac{t_{n}}{s_{n}}$$ equals

The sum of the coefficients of all odd exponets of $$\displaystyle x$$ in the product of $$\displaystyle (1-x +x^{2}-x^{3}+x^{4}+...-x^{49}+x^{50})\times(1+x+x^{2}+x^{3}+...+x^{50})$$ equals

If$$\displaystyle\left ( 1+x \right )^{n}= \sum_{r= 0}^{n}C_{r}x^{r}$$, then the value of $$C_{0}-C_{2}+C_{4}-C_{6}+C_{8}-C_{10}+...$$ equals

If $$\left ( 1+x \right )^{n}= \sum_{r= 0}^{n}C_{r}x^{r}$$ then the value of $$3C_{1}+7C_{2}+11C^{3}+....+\left ( 4n-1 \right )C_{n}$$ is

The Coefficient of $$x^{53}$$ in $$\sum_{m= 0}^{100}{C_{m}}^{100}\left ( x-3 \right )^{100-m}2^{m}$$ is

The number of terms with integral coefficient in the expansion of $$\left ( 17^{\dfrac 13} +35^{\dfrac 12}\right )^{300}$$ is