Combination with Restrictions

Q.

How many words, with or without meaning, can be made from the letters of the word MONDAY, assuming that no letter is repeated if:

(i) 4 letters are used at a time?

(ii) all letters are used at a time?

(iii) all letters are used but first letter is a vowel?

Q.

If the sides $AB,BC,$ and $CA$ of a triangle $ABC$ have $3$, $5$ and $6$ interior points respectively, then the total number of triangles that can be constructed using these points as vertices is equal to:

1. $360$

2. $240$

3. $333$

4. $364$

Q.

Eight chairs are numbered $1$ to $8$. Two women and three men wish to occupy one chair each. First, the women choose the chairs from amongst the chairs marked $1$ to $4$, and then the men select the chairs from amongst the remaining. The number of possible arrangements is:

1. $C_3^6\times C_2^4$

2. $P_2^4\times P_3^4$

3. $=P_2^4\times P_3^6$$C_2^4\times C_3^4$

4. None of these.

Q.

If 7 points out of 12 are in the straight line, then the numbers of triangles formed by joining them is

1. 201

2. 19

3. 185

4. None of these

Q.

How many 5 digit even numbers can be made from the digits 0, 1, 2, 3, 4, 5 if repetition is not allowed?

1. 360

2. 96

3. 120

4. 312

Q.

The number of triangles that can be formed by choosing the vertices from a set of $12$ points, seven of which lie on the same straight line, is?

1. $185$

2. $175$

3. $115$

4. $105$

Q.

If A = {x : x is a natural number}, B = {x : x is an even natural number}, C = {x : x is an odd natural number } and D = {x : x is a prime number}, find :

(i) $A\cap B$ (ii) $A\cap C$ (iii) $A\cap D$

(iv) $B\cap C$ (v) $B\cap D$ (vi) $C\cap D$

Q. A rectangle with sides 2m -1 and 2n -1 is divided into squares of unit length by drawing parallel lines as shows in the diagram, then the number of rectangles possible with odd side lengths is

1. $m(m+1)n(n+1)$
2. $4^{m+n-1}$
3. $(m+n-1{)}^2$
4. $m^2{n}^2$

Q.

The number of natural numbers less than $1000$ that are divisible by $5$ in which no digit occurs more than once in the same number is

1. $154$
2. $136$
3. $144$
4. $152$

Q.

How many numbers between 2000 and 3000 can be formed from the digits 2, 3, 4, 5, 6, 7 when repetition of digits is not allowed ?

Q. The number of numbers greater than or equal to 1000 but less than or equal to 4000 is formed using the digits 0, 1, 2, 3, 4 (repetition allowed) is

1. 376
2. 370
3. 360
4. 366

Q.

Simplify the following mathematical expressions :

$25\times 5-12+18÷9$

Q. A committee of $5$ men and $3$ women is to be formed out of $7$ men and $6$ women. If two particular women are not to be included together in the committee, then the number of committees that can be formed is

1. $420$
2. $540$
3. $336$
4. $216$

Q.

In a cricket board, Anil Jain is a cricket team selector. He can select a cricket team, 11 from 17 players in which only 5 players can bowl.

In how many ways exactly 4 bowlers must include out of 11 players?

Q.

There are four men and six women on the city council. If one council member is selected for a committee at random, how likely is it that it is a woman?

Q.

$A\left(z_1\right),B\left(z_2\right),C\left(z_3\right)$ are the vertices of a right angeled isosceles triangle ABC. If $\angle C$ = $\frac{\pi }{2}$, then:

1. 

2. 

3. 

4. 

Q.

A team of 4 students is to be sent for a competition. 12 students offered their services for the same . But from past experience, it was observed that 5 students who had offered the services were not true to their work and they did mischief one time or the other are not to be selected. In how many ways, can the 4 students be selected for a competition?

Q.

Mark three non-collinear points $\mathrm{A},\mathrm{B}\mathrm{and}\mathrm{C}$ in your notebook . Draw lines through these points taking two at a time and name these lines . How many such different lines can be drawn ?

Q.

Write the following sets in roster form:

(i) A = {x : x is an integer and -3 < x < 7}

(ii) B = {x : x is a natural number less than 6 }

(iii) C = {x : x is a two-digit natural number such that the sum of its digits is 8 }

(iv) D = {x : x is a prime number which is divisor of 60}

(v) E = The set of all letters in the word TRIGONOMETREY

(vi) F = The set of all letters in the word BETTER

Q. There are 25 railway stations between Nellore and Hyderabad. The number of different kinds of single second class tickets to be printed so as to enable a passenger to travel from one station to another is

1. ${}^{25}{\text{P}}_2$
2. ${}^{26}{\text{P}}_2$
3. ${}^{26}{\text{P}}_2$
4. ${}^{27}{\text{P}}_2$

Q. How many elements has $P\left(A\right)$ , if $A=\varphi$?

Q. If one quarter of all the subsets (containing three elements) of the set $A=\{a_1,a_2,a_3,\dots ,a_n\}$ contains the element $a_3,$ then the value of $n$ is

1. $10$
2. $12$
3. $14$
4. None of these

Q. The number of natural numbers less than 1000 that are divisible by 5 in which no digit occurs more than once in the same number is

1. 154
2. 136
3. 144
4. 152

Q.

How many numbers lying between 10 and 1000 can be formed from the

digits 1, 2, 3, 4, 5, 6, 7, 8, 9 (repetition is allowed)

1. 810

2. 1024

3. 2346

4. 729

Q. The number of different seven digit numbers that can be written using only the three digits $1,2$ and $3$ with the condition that the digit $2$ occurs twice in each number is

1. ${}^7{P}_2\times 2^5$
2. ${}^7{C}_2\times 2^5$
3. ${}^7{C}_2\times 5^2$
4. None of these

Q.

A is a set with 6 elements. So, the number of subsets is:

1. 24

2. 12

3. 64

4. 32

Q. Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then, the value of $\frac{m}{n}$ ?___

Q. The number of ways that 5 prizes be distributed among 4 boys while each boy is eligible for any number of prizes is

1. 10
2. 
3. 
4. 20

Q. The number of distinct straight-lines formed from $5$ non-collinear points is

1. $24$
2. $5$
3. $10$
4. $16$

Q.

There are $\mathrm{n}$ straight lines in a plane, no two of which are parallel and no three passes through the same point, Their points of intersection are joined. Then the number of fresh lines thus obtained is?

1. $\frac{\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)}{8}$

2. $\frac{\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)(\mathrm{n}-3)}{6}$

3. $\frac{\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)(\mathrm{n}-3)}{8}$

4. None of the above.