nCr Definitions and Properties

Q. (1) The sum of all the values of $r$ satisfying ${}^{39}{C}_{3r-1}{-}^{39}{C}_{r^2}{=}^{39}{C}_{r^2-1}{-}^{39}{C}_{3r}\text{is}{\alpha }_1.$

(2) If ${}^{2n+3}{C}_{2n}{-}^{2n+2}{C}_{2n-1}=15.(2n+1)$ then the value of $n$ is ${\alpha }_2.$

(3) If ${}^{56}{P}_{r+6}{:}^{54}{P}_{r+3}=30800:1$ then value of $r$ is ${\alpha }_3.$

(4) ${}^{n+2}{C}_8{:}^{n-2}{P}_4=57:16$ then the value of $n$ is ${\alpha }_4.$

$\begin{array}{cccc}& \text{List-I}& & \text{List-II}\\ \left(I\right)& \text{The value of}{\alpha }_1\text{is}& \left(P\right)& 41\\ \left(II\right)& \text{The value of}{\alpha }_2\text{is}& \left(Q\right)& 8\\ \left(III\right)& \text{The value of}{\alpha }_3\text{is}& \left(R\right)& 14\\ \left(IV\right)& \text{The value of}{\alpha }_4\text{is}& \left(S\right)& 19\end{array}$

$\text{Which of the following is only CORRECT Combination?}$

1. $\left(I\right)\to \left(R\right)$
2. $\left(II\right)\to \left(R\right)$
3. $\left(III\right)\to \left(S\right)$
4. $\left(IV\right)\to \left(Q\right)$

Q. Let $a_1,a_2,\dots ,a_{30}$ be in A.P., $S=\sum _{i=1}^{30}{a}_i$ and $T=\sum _{i=1}^{15}{a}_{2i-1}$. If $a_5=27$ and $S-2T=75$, then the value of $a_{10}$ is

1. $47$
2. $52$
3. $42$
4. $57$

Q.

If ${}^{56}{P}_{r+6}:{}^{54}{P}_{r+3}$ = 30800:1, then r =

1. 31

2. 41

3. 51

4. 40

Q. The value of $1+\frac{4}{7}+\frac{9}{7^2}+\frac{16}{7^3}+\frac{25}{7^4}+\cdots \text{upto}\infty$ is

1. $\frac{27}{14}$
2. $\frac{21}{13}$
3. $\frac{49}{27}$
4. $\frac{256}{147}$

Q. The value of ${\sum }_{r=1}^{10}r{.}^r{P}_r$ =

1. 11!
2. 11! - 1
3. 11! + 1
4. 11! - 11

Q. The equation whose roots are the values of $r$ satisfying the equation ${}^{69}{C}_{3r-1}-{}^{69}{C}_{r^2}={}^{69}{C}_{r^2-1}-{}^{69}{C}_{3r}$ is

1. $x^3-10x^2+21x=0$
2. $x^2+10x+21=0$
3. $x^2-10x+21=0$
4. $x^2-21x+10=0$

Q. A team of $12$ railway station masters is to be divided into two groups of $6$ each, one for day duty and the other for night duty. The number of ways in which this can be done if two specified persons $A$ and $B$ should not be included in the same group is

1. $420$
2. $504$
3. $714$
4. $924$

Q. The number of ways of selecting two squares from a chess board so that they have exactly one common corner is

1. $36$
2. $72$
3. $98$
4. $112$

Q. The number of ordered pairs $(r,k)$ for which $6{\cdot }^{35}{C}_r=(k^2-3){\cdot }^{36}{C}_{r+1}$, where $k$ is an integer, is :

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

Q. Number of points of intersection of diagonals of polygon with $2009$ sides which are situated inside the polygon.

1. ${}^{2009}{C}_3$
2. ${}^{2009}{C}_4$
3. $2\times {}^{2009}{C}_2$
4. ${}^{2009}{C}_2$

Q. If P(n, 5) = 20P (n, 3), then n =

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

Q. The number of ways of selecting two squares on a chess board such that they have a side in common is

1. $228$
2. $112$
3. $108$
4. $110$

Q. If P (10, r) = 5040, then r =

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

Q. The value of $\sum _{r=0}^n(-1{)}^r\frac{{}^n{C}_r}{{}^{r+2}{C}_r}$ is equal to

1. $\frac{2}{n+1}$
2. $\frac{2}{n-1}$
3. $\frac{2}{n+2}$
4. $\frac{2}{n-2}$

Q.

For all integers $a$ and $b$, ${(a+b)}^2\ge a^2+b^2$ and ${(a-b)}^2\le a^2+b^2$.

1. True
2. False

Q. If ${}^n{C}_{r-1}=36,{}^n{C}_r=84and{}^n{C}_{r+1}=126$ then r is

1. 1
2. 2
3. 3
4. none of these

Q. If ${}^n{\text{C}}_4,{}^n{\text{C}}_5$ and ${}^n{\text{C}}_6$ are in A.P., then $n$ can be :

1. $9$
2. $11$
3. $12$
4. $14$

Q. Number of ways of arranging $5$ different objects in the squares of given figure in such a way that no row remains empty and one square can't have more then one object.

1. $44\times 5!$
2. $48\times 5!$
3. $36\times 5!$
4. $40\times 5!$

Q. If $C_r={}^{25}{C}_r$ and $C_0+5\cdot C_1+9\cdot C_2+\cdots +101\cdot C_{25}=2^{25}\cdot k$ $k$ is equal to

Q. Let $T_n$ be the number of all possible triangles formed by joining vertices of $n$-sided regular polygon. If $T_{n+1}-T_n=10$, then the value of $n$ is :

1. $7$
2. $5$
3. $10$
4. $8$