Multinomial Theorem

Using matrices, what would be the value of $x, y, z$ for the following system of equations :

$\begin{array}{l}x+2y+z=7\end{array}$

$x+3z=11$

$2x-3y=1$

Coefficient of $x^2{y}^3{z}^4$ in ${\left(ax+by+cz\right)}^9$ 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 number of distinct terms in the expansion of ${\left(x^4+\frac{1}{x^4}+1\right)}^{100}$ 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

Let ${\left(1+x+x^2\right)}^{25}=a_0+a_{1}x+a_2{x}^2+\dots \dots +a_{50}{x}^{50}.$ If $A=\left(-a_1+a_3-a_5+a_7-a_9+\dots \dots -a_{49}\right),$, then find $\frac{A}{8}:-$

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 :-

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

The coefficient of $x^7$ in $1+\left(1+x\right)+{\left(1+x\right)}^2+\dots \dots +{\left(1+x\right)}^{100}$ is

Coefficient of $x^3$ in the expansion of $\sum _{i,j,k,\ell \in \left\{1,2\right\}}\left(1+x^{\mathit{i}}\right)\left(1+x^{\mathit{j}}\right)\left(1+x^{\mathit{k}}\right)\left(1+{\mathrm{x}}^{\ell }\right)$ is

The number of integral solutions of the equation $x+y+z+w=1$ such that $x,y,z,w>-4$, is equal to

Total number of points having position vector $a\widehat{i}+b\widehat{j}+c\widehat{k}$  and satisfing the condition $6\le a+b+c\le 10$ where $(a,b,c\in N)$ is