nCr Definitions and Properties

Q.

Given $n\left(U\right)=20,n\left(A\right)=12,n\left(B\right)=9,n(A\cap B)=4$, where $\cup$ is the universal set, $A$ and $B$ are subset of $\cup$, then $n(A\cup B^c)=$

1. $17$

2. $9$

3. $11$

4. $3$

5. $16$

Q.

If $n\left(A\right)=3,n\left(B\right)=5$and $n(A\cap B)=2$ then $n\left[\right(A\times B)\cap (B\times A\left)\right]=$

1. $5$

2. $3$

3. $4$

4. $6$

Q. In how many ways can a committee be formed of 5 members from 6 men and 4 women if the committee has at least one woman

1. 186
2. 246
3. 252
4. None of these

Q. The number of ways in which $5$ different balls can be placed in $3$ identical boxes such that no box remains empty is

Q. If $6^{83}+8^{83}$ is divided by $49,$ then the remainder is

1. $35$
2. $5$
3. $1$
4. $0$

Q.

if $A=\left\{4^n-3n-1:nbelongstoN\right\}$ and $B=\left\{9(n-1):nbelongstoN\right\}$, then

1. $B\subset A$

2. $A\cup B=N$

3. $A\subset B$

4. None of these

Q.

If polynomials $a{x}^3+3x^2-3$ and $2x^3-5x+a$ when divided by $(x-4)$ leave the remainders as $R_1$ and $R_2$ respectively.

Find the values of $a$ in $2R_1-R_2=0$

Q. If A =[i 0] B=[0 i] , where i=square root of [0 -i] [i 0] Minus one, then the correct relation is 1.A+B=0 2.A square=B square 3.A-B=0 4.Asquare+B square=0

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. $8$

Q. Let f(x)=tanx, g(x)=square root of (1-x^2) then g(f(x)) is

Q.

Simplify and express each of the following in exponential form:${25}^4÷5^3$

Q.

If$a,bandc$ are three unequal numbers such that $a,bandc$are in A.P and $b-a,c-banda$are in G.P., then $a:b:c$ is

1. $1:2:3$

2. $2:3:1$

3. $1:3:2$

4. $3:2:1$

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.

If $[a\times bb\times cc\times a]=\lambda {[abc]}^2$, then $\lambda$is equal to

1. $0$

2. $1$

3. $2$

4. $3$

Q. A natural number is selected at random from $1$ to $1000.$ Then the probability that non-zero digits appear at most once is

1. $\frac{11}{500}$
2. $\frac{7}{250}$
3. $\frac{4}{125}$
4. $\frac{17}{750}$

Q.

If ${}^{2n}{C}_3:{}^n{C}_2=44:3$, then for which of the following values of $r$, the value of ${}^n{C}_r$ will be $15$?

1. $r=3$

2. $r=4$

3. $r=6$

4. $r=5$

Q. If $\sum _{i=1}^{20}{\left(\frac{{}^{20}{C}_{i-1}}{{}^{20}{C}_i+{}^{20}{C}_{i-1}}\right)}^3=\frac{k}{21},$ then $k$ equals :

1. $50$
2. $100$
3. $200$
4. $400$

Q.

A student has to answer 10 questions, choosing at least 4 from each of part A and part B. If there are 6 questions in part A and 7 in part B, in how many ways can the student choose 10 questions ?

Q.

If ${}^{n+2}{C}_8:{}^{n-2}{P}_4=\frac{57}{16}$ then $n$ is equal to ?

1. $19$

2. $2$

3. $20$

4. $5$

Q.

Four couples (husband and wife) decided. Find the number of different committees that can be formed in which no couple finds a place.

1. $12$

2. $14$

3. $16$

4. $24$

Q. The number of 4 digit numbers that can be formed with 0, 1, 2, 3, 5 which are divisible by 2 or 5 is

1. 84
2. 60
3. 53
4. 120

Q.

If ${}^{43}{C}_{r-6}{=}^{43}{C}_{3r+1},$ then the value of r is

1. 12

2. 8

3. 6

4. 10

5. 14

Q.

If the letters of the word ‘SACHIN’ are arranged in all possible ways and these words are written in dictionary order, then the word "SACHIN’ appears at serial number?

1. $600$

2. $601$

3. $602$

4. $603$

Q.

If ${}^{n}C_r$ denotes the number of combinations of $n$ things taken $r$ at a time, then the expression ${}^{n}C_{r+1}+{}^{n}C_{r-1}+2.{}^{n}C_r$ equals?

1. ${}^{n+2}C_r$

2. ${}^{n+2}C_{r+1}$

3. ${}^{n+1}C_r$

4. ${}^{n+1}C_{r+1}$

Q. Let a cricket player played $n(n>1)$ matches during his career. If $T_r$ represents the runs made by the player in $r^{\text{th}}$ match such that $T_1=6$ and $T_r=3T_{r–1}+6^r$ for $2\le r\le n$, then the runs scored by him in $100$ matches is

1. $\frac{3}{5}[4\cdot 6^{100}-5\cdot 3^{100}-1]$
2. $\frac{2}{5}[4\cdot 6^{100}-5\cdot 3^{100}+1]$
3. $\frac{3}{5}[4\cdot 6^{99}-5\cdot 3^{99}+1]$
4. $\frac{3}{5}[4\cdot 6^{100}-5\cdot 3^{100}+1]$

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

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

Q.

In an examination, a student has to answer 4 questions out of 5 questions ; questions 1 and 2 are however compulsory, Determine the number of ways in which the student can make the choice.

Q.

The number which should be added to the numbers $2,14,62$ so that the resulting numbers may be in G.P., is

1. $1$

2. $2$

3. $3$

4. $4$

Q.

In a $12$-storey house ten people enter a lift cabin. It is known that they will leave the lift in pre-decided groups of $2,3,$ and $5$ people at different storeys. The number of ways they can do so if the lift does not stop at the second storey is

1. $78$

2. $112$

3. $720$

4. $132$

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. $110$