Multinomial Expansion

Q.

How to prove ${{}^{n}C}_r+{{}^{n}C}_{r-1}={{}^{n+1}C}_r$

Q.

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

1. 11

2. 33

3. 66

4. None

Q.

A window is in the form of a rectangle surmounted by a semi-circle opening. The perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

Q. How many triplets of non-negative integers $(x,y,z)$ satisfy the equation $xyz+xy+yz+zx+x+y+z=2012$?
(correct answer + 3, wrong answer 0)

Q. The number of elements in the set $\{n\in \{1,2,3,...,100\}|(11{)}^n>(10{)}^n+(9{)}^n\}$ is

Q. The coefficient of $x^4$ in the expansion of $(1+x+x^2{)}^{10}$ is

Q. Find the pairs of equal sets, if any, give reasons:
$A=\left\{0\right\},B=\{x:x>15\text{and}x<5\}$,
$C=\{x:x-5=0\},D=\{x:x^2=25\}$
$E=\{x:x$ is an integral positive root of the equation of $x^2-2x-15=0\}$

Q.

If number of terms in the expansion of $(x-2y+3z{)}^n$ are 45, then n =

1. 7

2. 8

3. 9

4. None of these

Q. The coefficient of $x^2$ in the expansion of the product $(2–x^2)\left[\right(1+2x+3x^2{)}^6+(1–4x^2{)}^6]$ is

1. $107$
2. $155$
3. $108$
4. $106$

Q. The number of distinct terms in the expansion of ${(x+\frac{1}{x}+x^2+\frac{1}{x^2})}^{15}$ is
(Different power of $x$ means different terms)

1. $60$
2. $61$
3. $30$
4. $31$

Q.

The sum of the infinite series$\frac{2^2}{2!}+\frac{2^4}{4!}+\frac{2^6}{6!}+\dots$ is equal to

1. $\frac{(e^2+1)}{2e}$

2. $\frac{(e^4+1)}{2e^2}$

3. $\frac{{(e^2-1)}^2}{2e^2}$

4. None of these

Q. The coefficient of $x^{10}$ in the expansion of $[1+x^2(1-x){]}^8$ is

1. $400$
2. $576$
3. $470$
4. $476$

Q.

Find a positive value of m for which the coefficient of x² in the expansion

(1 + x)^m is 6.

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

Q. The coefficient of $x^{28}$ in the expansion of $(1+x^3-x^6{)}^{30}$ is

1. ${}^{30}{C}_6$
2. ${}^{30}{C}_3$
3. $1$
4. $0$

Q. The largest term in the expansion of $(3+2x{)}^{50}$ where $x=\frac{1}{5}$ is

1. 
2. 
3. 
4. 

Q. The constant term in the expansion of ${(x^2-\frac{1}{x^2})}^{16}$ is

1. ${}^{16}{C}_8$
2. ${}^{16}{C}_7$
3. ${}^{16}{C}_9$
4. ${}^{16}{C}_{10}$

Q. If the number of distinct terms in expansion of ${(x+y+z+\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx})}^2$ and ${(x+y+z+\frac{1}{x}+\frac{1}{y}+\frac{1}{z})}^2$ is $m$ and $n$ respectively, then the value of $m+n$ is

Q. If the sum of the coefficients in the expansion
${({\alpha }^2{x}^2-2\alpha x+1)}^{51}$
vanishes, then the value of $\alpha$ is ____

1. 1
2. 2
3. -1
4. -2

Q. The coefficient of $a^8{b}^4{c}^9{d}^9$ in $(abc+abd+acd+bcd{)}^{10}$ is

1. $10!$
2. $1260$
3. $5040$
4. $2520$

Q.

$(1.2{)}^{4000}$ is larger than 800

1. True

2. False

Q. The number of positive integral solutions of the equation det$\left[\begin{array}{ccc}{y}^3+1& y^{2}z& y^{2}x\\ y{z}^2& z^3+1& z^2\\ y{x}^2& x^z& x^3+1\end{array}\right]=11is$

1. 1
2. 2
3. 3
4. 4

Q.

In the expansion of $(1+x{)}^n(1+y{)}^n(1+z{)}^n$, the sum of the coefficients of the terms of degree r is

1. 

2. 

3. 

4. 

Q. The number of ordered triplets $(a,b,c)$ where $a,b\in W$ and $c\in N$ such that origin and the point $(1,1,4)$ lie on the same side of the plane $ax+by+cz=39$ is

1. $2280$
2. $2190$
3. $2180$
4. $2290$

Q. The coefficient of $x^2{y}^3$ in the expansion of $(1-x+y{)}^{20}$ is

1. $\frac{20!}{2!3!}$
2. $-\frac{20!}{2!3!}$
3. $\frac{20!}{(15!)(3!)(2!)}$
4. $\frac{20!}{5!2!3!}$

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

Q. Coefficient of $a^5{b}^4{c}^7$ in the expansion of $(bc+ca+ab{)}^8$ is _____

1. 180
2. 280
3. 360
4. 300

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. If the number of terms in the expansion of $(x+y+z{)}^n$ is $36$, then the value of $n$ is

Q. If the coefficient of $x^{1274}$ in the expansion of $(x+1)(x-2{)}^2(x+3{)}^3(x-4{)}^4\cdots \cdots (x+49{)}^{49}(x-50{)}^{50}$ is $-k$, then the value of $k$ is