Multinomial Theorem

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

Let ${\left(1+x+x^2\right)}^{2014}=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\dots +a_{4028}{x}^{4028},$ and let

$A=a_0-a_3+a_6-\dots +a_{4026},$

$B=a_1-a_4+a_7-\dots -a_{4027},$

$C=a_2-a_5+a_8-\dots +a_{4028},$

Then-

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

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

If $a,\mathrm{ }\mathrm{b},\mathrm{ }\mathrm{c},d\mathrm{ }$ are odd natural numbers such that $a+b+c+d=20$ then the number of values of $a, b, c, d,$ is

The number of ordered triplets of positives integers which are solutions of the equation $x+y+z=100$, is

In how many ways can Rs 16 be divided into 4 persons when none of them get less than Rs 3?

The number of terms in the expansion of ${\left(a+b+c\right)}^n$ will be

If $A={1000}^{1000}$ and $B={\left(1001\right)}^{999}$, then

The coefficient of $a^3{b}^{4}c$ in the expansion of ${(1+a-b+c)}^9$is equal to

In the expansion of ${(1+3x+2x}^2{)}^6$ the coefficient of $x^{11}$ is

If $a_k$ is the coefficient of${ x}^k$ in the expansion of ${(1+x+x^2)}^n$ for $k=0, \mathrm{1,2},$...., $2n$ then $a_1+2a_2+3a_3+...+2n{a}_{2n}=$

If the sum of the coefficient in the expansion of $(a^2{x}^2-6ax+11{)}^{10}$, where $a$ is constant is $1024$, then the value of $a$ is

The coefficient of $a^5{b}^6{c}^7$ in the expansion of ${(bc+ca+ab)}^9$ is

If ${(1+x-{3x}^2)}^{10}=1+a_{1}x+a_2{x}^2+...+a_{20}{x}^{20},$ then $a_2+a_4+a_6+...+a_{20}$is equal to

The coefficient of $a^3{b}^{4}c$ in the expansion of ${(1+a-b+c)}^9$is equal to

If ${(1-x+x^2)}^n=a_0+a_{1}x+$.....$a_{2n}{x}^{2n}$ then the value of $a_0+a_2+a_4+$....$+a_{2n}$ is