Multinomial Theorem

Let ${\left(a+bx+c{x}^2\right)}^{10}=\sum _{i=10}^{20}{p}_i{x}^i, a, b, c\in \mathrm{\mathbb{N}}$. If $p_1=20$ and $p_2=210$, then $2\left(a+b+c\right)$ is equal to 

The coefficient of $x^7$ in ${\left(1-x+2x^3\right)}^{10}$ is __________ .

The number of ways of giving $20$ distinct oranges to $3$ children such that each child gets at least one orange is $\_\_\_\_\_$

The coefficient of $x^7$ in ${\left(1-2x+x^3\right)}^{10}$ is 

If $\frac{3}{4!}+\frac{4}{5!}+\frac{5}{6!}+...+50 \mathrm{terms}=\frac{1}{3!}-\frac{1}{\left(k+3\right){\displaystyle !}}$, then sum of coefficients in the expansion of ${\left(1+2x_1+3x_2+...+100x_{100}\right)}^k$ is

(where, $x_1,x_2,x_3,...,x_{100}$ are independent variables)

If ${\left(1+x-2x^2\right)}^4=1+a_{1}x+a_2{x}^2+...+a_8{x}^8$, then the expression $a_2+a_4+a_6+a_8$ is equal to

If the number of terms in ${\left(x+1+\frac{1}{x}\right)}^n$, $n$ being a natural number is $301$ then $n=$

The number of rational terms in ${\left(\sqrt{2}+\sqrt[3]{3}+\sqrt[6]{5}\right)}^{10}$ is

If $\lambda$ is the coefficient of $x^5$ in the expansion of ${\left(x^2-x-1\right)}^5$, then the value of $\lambda +12$ is

Find the number of terms in ${\left(a+b+c+d+e\right)}^{15}$.

Find the number of terms in ${\left(a_1+a_2+a_3+\cdots +a_{20}\right)}^5$.

Find the number of terms in ${\left(a+b+c+d+e+f+g\right)}^{10}$.

Find the number of terms in ${\left(a+b+c+d\right)}^{12}$.

Find the number of terms in ${\left(a+b+c\right)}^{20}$.

The sum of the coefficient of all the terms in the expansion of $(2x-y+z{)}^{20}$ in which $y$ do not appear at all while $x$ appears in even powers and $z$ appears in odd powers is -

The coefficient of $a^8{b}^4{c}^9{d}^9$ in $(\alpha bc+\alpha bd+acd+bcd{)}^{10}$ is

Let ${\left(1-2x+3x^2\right)}^{10}=a_0+a_{1}x+a_2{x}^2+\dots ..+a_n{x}^n,a_n\ne 0$ , then the arithmetic mean of $a_0, a_1,a_2,\dots \dots a_n$ is

The digit at hundredths place in coefficient of $x^{22}$ in the expansion of ${\left(1-x^7+x^8\right)}^{25}$ is equal to

The coefficient of $a^8{b}^4{c}^9{d}^9$ in ${\left\{ab\left(c+d\right)+cd\left(a+b\right)\right\}}^{10}$ is

Number of Irrational terms in the Expansion of ${\left(2^{\frac{1}{3}}+3^{\frac{1}{2}}+5^{\frac{1}{6}}\right)}^{10}$ is equal to :-