Permutations and Combinations

Q. Let $X=\{1,2,3,4,5\}$. The number of different ordered pairs $(Y,Z)$ that can be formed such that $Y\subseteq X,Z\subseteq X$ and $Y\cap Z$ is empty, is

1. $5^2$
2. $3^5$
3. $5^3$
4. $2^5$

Q. Which of the following expression(s) represent a polynomial

1. $2x^3-3x^2+5$
2. $x-5$
3. $x^2-5\sqrt{x}+2$
4. $5x^5-4x^4+\frac{3}{x^3}-2x^2+x+4$

Q.

There are 3 letters and 3 directed envelopes. Write the number of ways in which no letter is put in the correct envelope.

Q. Let$I\left(n\right)=n^n,J\left(n\right)=\mathrm{1.3.5....}(2n-1)\text{for all}(n>1),n\in N,$ then

1. $I\left(n\right)>J\left(n\right)$
2. $I\left(n\right)<J\left(n\right)$
3. $I\left(n\right)\ne J\left(n\right)$
4. $I\left(n\right)=\frac{1}{2}J\left(n\right)$

Q. The number of irrational terms in the expansion of ${(\sqrt[8]{85}+\sqrt[6]{62})}^{100}$ is

Q. The fourth power of the common difference of an A.P with integer entries is added to the product of any four consecutive terms of it, resulting sum is

1. 
2. 
3. 
4. 

Q. Let $T_n=(n^2+1)n!$ and $S_n=T_1+T_2+T_3+\cdots +T_n.$ If $\frac{T_{10}}{S_{10}}=\frac{a}{b},$ where $a$ and $b$ are relatively prime natural numbers, then the value of $(b-a)$ is

1. $2$
2. $9$
3. $14$
4. $13$

Q. The maximum value of ${}^{2n}{C}_r$ is equal to

1. ${}^{2n-1}{C}_r$
2. ${}^{2n-1}{C}_{r-1}$
3. $2\left({}^{2n}{C}_r\right)$
4. $2\left({}^{2n-1}{C}_r\right)$

Q.

Is $x^3-\frac{1}{x{a}^3}$ a polynomial or not?

Q. The total number of terms in the expansion of $(x+a{)}^{47}-(x-a{)}^{47}$ after simplification is

1. $24$
2. $47$
3. $48$
4. $96$

Q. If $\int \frac{x^{2018}}{1+x+\frac{x^2}{2!}+\cdots +\frac{x^{2018}}{2018!}}dx=m!x-m!\ln |P\left(x\right)|+C$ for arbitrary constant of integration $C,$ then

1. $m=2019$
2. $m=2018$
3. $P\left(x\right)=1+x+\frac{x^2}{2!}+\cdots +\frac{x^{2018}}{2018!}$
4. $P\left(x\right)=x+\frac{x^2}{2!}+\cdots +\frac{x^{m+1}}{(m+1)!}$

Q. If set $A=\left\{\right(r,s)|r,s\in W\}$, then the number of element(s) in set $A$ such that ${}^5{C}_r\cdot {}^6{C}_s=1$ is

1. $0$
2. $2$
3. $3$
4. $4$

Q. The number of terms in the expansion of $(4x^2-16x+16{)}^5$ is

Q. If set $A=\left\{\right(r,s)|r<s;r,s\in W\}$, then the number of element(s) in set $A$ such that ${}^7{C}_r+{}^7{C}_{r-1}={}^8{C}_s$ is

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

Q.

Let $X=\{1,2,3,4,5\}$. The number of different ordered pairs $(Y,Z)$ that can be formed such that $Y\subseteq X,Z\subseteq X$ and $Y\cap Z$ is empty, is

1. $5^2$

2. $3^5$

3. $2^5$

4. $5^3$

Q. The total number of terms in the expansion of ${\left((1+x^{2/3})(1+x^{4/3}-x^{2/3})\right)}^{2021}$ is

1. $4043$
2. $3033$
3. $2022$
4. $2021$

Q. If ${}^5{C}_r=r\cdot {}^5{C}_{r-1}$, then the number of value(s) of $r$ is

1. $2$
2. $1$
3. $3$
4. $0$

Q. The number of terms in the expansion of $(1-3x{)}^{18}$ is

1. $18$
2. $19$
3. $20$
4. $17$

Q. The number of solution(s) for $\text{sgn}(x+1)=2x^2-x$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q. Which of the following expression(s) represent a polynomial

1. $2x^3-3x^2+5$
2. $x-5$
3. $x^2-5\sqrt{x}+2$
4. $5x^5-4x^4+\frac{3}{x^3}-2x^2+x+4$

Q. If $p\left(n\right):n^2<2^n$ is true for $n\in N-\left\{1\right\}.$ then the minimum value of $n=$
(use principle of mathematical induction)

Q. If $\sum _{r=1}^{k}r=\frac{1}{2}(n^2+11n+30)$ is true for $n\in N,$ then $k=$
(use principle of mathematical induction)

1. $n+7$
2. $n+6$
3. $n+4$
4. $n+5$

Q. In the expansion of ${(\sqrt{2}+\sqrt[5]{53})}^{120}$ the number of irrational terms is

1. $54$
2. $12$
3. $13$
4. $108$

Q. In the following question, the numbers/letters are arranged based on some pattern or principle.Choose the correct answer for the term marked by the symbol (?)

1. $86$
2. $138$
3. $76$
4. $120$

Q.

If the sum of frist n terms of an A.P. be equal to the sum of its

first m terms, (m ≠ n), then the sum of its first (m+n) terms

will be

1. 0

2. n

3. m

4. m+n

Q. Which of the following expression(s) represent a polynomial

1. $2x^3-3x^2+5$
2. $x-5$
3. $x^2-5\sqrt{x}+2$
4. $5x^5-4x^4+\frac{3}{x^3}-2x^2+x+4$

Q. If total $180n,n\in N$ different matrices can be formed by using all the roots of equation $(x-1{)}^2(x-2{)}^3(x-3{)}^4=0.$ Then the value of $n$ is

Q. The number of words that can be formed by the letters of $MATHS$ is

Q. Let $S=\{1,2,3,4\}$. The total number of unordered pairs of disjoint subsets of $S$ is equal to

1. $41$
2. $42$
3. $25$
4. $34$

Q.

Is $p^2+3p-\frac{2}{p^2}$ polynomial or not?