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 ${\left(a+b+c\right)}^{20}$.

Total number of $4$ letters words that can be formed with the letters of the word $"RATIONAL"$ is

Number of positive integral solutions of $(x_1+x_2+x_3)(y_1+y_2)=55,$ is

The sum of the rational term(s) in the expansion of $(\sqrt{3}+\sqrt[3]{2}+\sqrt[4]{5}{)}^8$ is

There are $10$ seats in the first row of a theatre of which $4$ are to be occupied. The number of ways of arranging $4$ persons so that no two persons sit side by side is:

The number of distinct terms in the expansion of $(x+2y-3z+5w-7u{)}^n$ is:

The coefficient of $x^4$ in the expansion of ${\left(1-x+x^2-x^3\right)}^4$ is

Coefficient of $t^{24}$ in ${\left(1+t^2\right)}^{12}\left(1+t^{12}\right)(1+t^{24})$ is

If in the expansion of ${\left(1+x\right)}^m{\left(1-x\right)}^n,$ the coefficient of $x$ and $x^2$ are$3$ and $- 6$ respectively, then $m$ is

The numbers of terms in the expansion of ${\left(a^3+\frac{1}{a^3}+1\right)}^{100}$ is

The number of distinct terms in the expansion of ${\left(x+2y-3z+5w-7u\right)}^n$ is

The coefficients of $x^2{y}^2, yz{t}^2$ and xyzt in the expansion of ${\left(x+y+z+t\right)}^4$ are in the ratio

Coefficient of $a^{8}bc$ in the expansion of ${\left(a+b+c\right)}^{10}$ is