Multinomial Expansion

Q. Expand the following:
${(x+\frac{1}{x})}^6$

Q. Find a, b and n in the expansion of $(a+b{)}^n$. If the first three terms of the expansion are 729, 7290 and 30375 respectively.

Q.

Find the $4^{th}$ term in the expansion of $(x-2y{)}^{12}$.

Q. Find the 4th term from the end in the expansion of ${(\frac{3}{x^2}-\frac{x^3}{6})}^7$.

Q. Write the general term in the expansion of $(x^2-y{)}^6$.

Q.

If the 6th, 7th and 8th terms in the expansion of $(x+a{)}^n$ are respectively 112, 7 and , $\frac{1}{4}$, find x, a and n.

Q. Expand the following:
${(\frac{x}{3}+\frac{1}{x})}^5$

Q. Find the 13th term in the expansion of ${[9x-\frac{1}{3\sqrt{x}}]}^{18},x\ne 0$.

Q.

The value of \((1+i)^2 × (1- i)^2 is

1. 8i

2. 8

3. -8

4. 32

Q. Sum of coefficients of the following expansion $(x+2y+3z{)}^8$ is ____

1. $3^9$
2. $6^8$
3. $3^8$
4. $6^9$

Q.

The sum of coefficients of integral powers of x in the binomial expansion of (1 - 2√x)⁵⁰ is :

1. 

2. 

3. 

Q. If the number of terms in the expansion of ${(1-\frac{2}{x}+\frac{4}{x^2})}^n,x\ne 0$, is 28, then the sum of the coefficients of all the terms in this expansions, is

1. 64
2. 2187
3. 243
4. 729

Q.

In the expansion of $(1+ax{)}^n$, the first three terms are $1+12x+64x^2$, then n=

1. 16

2. 18

3. 20

4. 9

Q. Find the value of ${}^{2n+1}{C}_0{+}^{2n+1}{C}_1{+}^{2n+1}{C}_2+......{+}^{2n+1}{C}_n{+}^{2n+1}{C}_{n+1}{+}^{2n+1}$$C_{n+2}+.....{+}^{2n+1}{C}_{2n+1}$ if $n=3$.
___

Q. $\mathrm{Number}\mathrm{of}\mathrm{terms}\mathrm{in}\mathrm{the}\mathrm{expansion}\mathrm{of}{\left(x_1+x_2+x_3+...+x_r\right)}^n\mathrm{is}\mathrm{equal}\mathrm{to}n+r-1Cr-1$

1. False
2. True

Q. The number of positive integral solutions of the equation $x+y+z+\omega =15$, where $x>1,y>2,z>3$ and $\omega <4$ is

Q. Consider the equation $x+y+z=16$ where $x,y,z>0$. Let $m$ and $n$ denote the number of positive integral solutions of the equation when $x$ is even and $y$ is odd, respectively. Then the value of $m+n$ is

1. $49$
2. $77$
3. $105$
4. $93$

Q.

$C_0-C_1+C_2-C_3+........+(-1{)}^n{C}_n$ is equal to

1. 

2. 

3. 

4. 0

Q. The number of terms in the expansion of $(x+y+z{)}^{10}$ is

1. $88$
2. $55$
3. $66$
4. $77$

Q.

${}^n{C}_1$ + ${}^n{C}_3$ + ${}^n{C}_5$..............=

1. 0

2. 1

3. 

4. 

Q. The coefficient of $\frac{1}{x}$ in the expansion of ${\left(1+x\right)}^n{\left(1+\frac{1}{x}\right)}^n$ is
(a) $\frac{n!}{\left[\left(n-1\right)!\left(n+1\right)!\right]}$

(b) $\frac{\left(2n\right)!}{\left[\left(n-1\right)!\left(n+1\right)!\right]}$

(c) $\frac{\left(2n\right)!}{\left(2n-1\right)!\left(2n+1\right)!}$

(d) none of these

Q.

Find the value of ${}^{n+1}{C}_1$ - ${}^{n+1}{C}_2$ + ${}^{n+1}{C}_3$ ..................$(-1{)}^{n+1}$ ${}^{n+1}{C}_{n+1}$

___

Q.

Find the value of $\sum _{r=1}^n(-1{)}^{r+1}$${}^{n-1}{C}_{r-1}$

___

Q.

Find the sum of the powers of X and Y of ${17}^{th}$ term in the expansion of $(X+Y{)}^{21}$

__

Q. The coefficient of x⁵ in the expansion of ${\left(1+x\right)}^{21}+{\left(1+x\right)}^{22}+...+{\left(1+x\right)}^{30}$
(a) ⁵¹C₅
(b) ⁹C₅
(c) ³¹C₆ − ²¹C₆
(d) ³⁰C₅ + ²⁰C₅

Q.

If $(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+.......+C_n{x}^n$, then

$\frac{C_1}{C_0}$ + $\frac{2C_2}{C_1}$ + $\frac{3C_3}{C_2}$ + ........+ $\frac{n{C}_n}{C_{n-1}}$ =

1. 

2. 

3. 

4. 

Q. The coefficients of 5th, 6th and 7th terms in the expansion of (1 + x)ⁿ are in A.P., find n.

Q. If $x+y+z+w=20$ where no variable may exceed $10,$ then the number of positive integral solutions is

Q. $\text{In the expansion of}(x^3+2x^2+x+4{)}^{15},\text{the coefficient of}{x}^2\text{is not divisible by}$

1. $8$
2. $25$
3. $27$
4. $64$

Q. If the coefficients of x² and x³ in the expansion of (3 + ax)⁹ are the same, then the value of a is
(a) $-\frac{7}{9}$

(b) $-\frac{9}{7}$

(c) $\frac{7}{9}$

(d) $\frac{9}{7}$