Combination with Restrictions

Q. Words of length $10$ are formed using the letters $A,B,C,D,E,F,G,H,I,J$. Let $x$ be the number of such words where no letter is repeated; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, $\frac{y}{9x}=$

Q.

There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two of them is

1. 45

2. 40

3. 39

4. 38

Q.

In how many ways can one select a cricket team of eleven from 17 players in which only 5 persons can bowl if each cricket team of 11 must include exactly 4 bowlers ?

Q.

Write the number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines.

Q. The number of all $3\times 3$ matrices $A$, with entries from the set $\{-1,0,1\}$ such that the sum of the diagonal elements of $\left(A{A}^T\right)$ is $3$, is

Q.

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (i) no girl ?

(i) at least one boy and one girl ?

(iii) at least 3 girls ?

Q.

Total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants is equal to

1. 60

2. 120

3. 7200

4. None of these

Q. A father with $8$ children takes them $3$ at a time to the zoological garden, as often as he can without taking the same $3$ children together more than once. Then the number of times a particular child will not go to the zoological garden is

Q. The number of ways of selecting $15$ teams from $15$ men and $15$ women such that each team consists of a man and a woman, is

1. $14!$
2. $15!$
3. $(15!{)}^2$
4. $1960$

Q.

There are 12 points in a plane. The number of the straight lines joining any two of them when 3 of them are collinear, is

1. 65

2. 64

3. 62

4. 63

Q. Consider three boxes, each containing $10$ balls labelled $1,2,\cdots ,10.$ Suppose one ball is randomly drawn from each of the boxes. Denoted by $n_i,$ the label of the ball drawn from the $i^{th}$ box, $(i=1,2,3)$. Then, the number of ways in which the balls can be chosen such that $n_1<n_2<n_3$ is :

1. $120$
2. $240$
3. $82$
4. $164$

Q.

In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls ?

Q.

How many four digit natural numbers not exceeding $4321$ can be formed with the digits $1,2,3$ and $4$, if the digits can repeat?

Q.

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

1. 96

2. 120

3. 48

4. 24

Q.

In how many ways can a football team of 11 players be selected from 16 players ?

How many of these will (i) include 2 particular players ? (ii) exclude 2 particular players ?

Q. Consider the letters of the word MATHEMATICS. The possible number of words

1. when no two vowels are together is $\frac{7!}{2!2!}{}^8{C}_4\frac{4!}{2!}$
2. when both $M$'s are together and both $T$'s are together but both $A$'s are not together is $28\times 7!$
3. when all vowels are together is $\frac{8!4!}{2!2!2!}$
4. when all consonants are together is $\frac{5!7!}{2!2!2!}$

Q.

In how many ways can a lawn tennis mixed double be made tip from seven married couples if no husband and wife play in the same set?

Q. Three ladies have brought one child each for admission to a school. The principal wants to interview the six persons one by one subject to the condition that no mother is interviewed before her child. The number of ways in which interviews can be arranged, is

1. $36$
2. $90$
3. $6$
4. $72$

Q.

There are 13 players of cricket, out of which 4 are bowlers. In how many ways a team of eleven be selected from them so as to include at least two bowlers ?

1. 42

2. None of these

3. 78

4. 72

Q.

A team of $4$ students is to be selected from a total of $12$ students. The total number of ways in which the team can be selected such that two particular students refuse to be together and other two particular students wish to be together only is equal to

1. $182$
2. $210$
3. $226$
4. $280$

Q. A candidate is required to answer $6$ out of $12$ questions which are divided into two parts $A$ and $B$ each containing $6$ questions and he/she is not permitted to attempt more than $4$ questions from any part. In how many different ways can he/she make up his/her choice of $6$ questions?

1. $800$
2. $850$
3. $700$
4. $750$

Q.

Consider a rectangle ABCD having $5,7,6,9$points in the interior of the line segments$AB,CD,BC,andDA$ respectively. Let $\alpha$ be the number of triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals having these points from different sides as vertices. Then $(\beta –\alpha )$ is equal to:

1. $1890$

2. $795$

3. $717$

4. $1173$

Q. A debate club consists of $6$ girls and $4$ boys. A team of $4$ members is to be selected from this club including a selection of a captain (from among these four members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

1. $380$
2. $320$
3. $260$
4. $95$

Q.

There are n straight lines in a plane, no two of which are parallel and no three pass through the same point. Their points of intersection are joined. Then the number of fresh lines thus obtained is

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

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

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

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

Q. Let $f\left(n\right)$ denotes the number of different ways in which the positive integer $n$ can be expressed as the sum of $1^{\prime }s$ and $2^{\prime }s$. For example $f\left(4\right)=5$, since
$4=2+2=2+1+1=1+2+1=1+1+2=1+1+1+1$.

Then which of the following(s) is (are) CORRECT ?

1. $f\left(6\right)=13$
2. $f\left(f\right(6\left)\right)=377$
3. $f\left(f\right(6\left)\right)=370$
4. $f\left(6\right)=11$

Q. A regular polygon of $10$ sides is constructed. The number of ways in which $3$ vertices can be selected so that no two vertices are consecutive is

Q.

There is a set of $m$ parallel lines intersecting a set of another $n$ parallel lines in a plane. The number of parallelograms formed, is

1. ${}^{m-1}{C}_2\times {}^{n-1}C_2$

2. ${}^m{C}_2\times {}^{n}C_2$

3. ${}^{m-1}{C}_2\times {}^{n}C_2$

4. ${}^m{C}_2\times {}^{n-1}C_2$

Q.

Given 11 points, of which 5 lie on one circle, other than these 5 no 4 lie on one circle. Then the number of circles that can be drawn so that each contains at least 3 of the given points is

1. 216

2. 156

3. 172

4. None of these

Q. You are given $8$ balls of different colours (black, white, ...). The number of ways in which these balls can be arranged in a row so that the two balls of particular colour (say red and white) may never come together, is

1. $8!-2\times 7!$
2. $6\times 7!$
3. $2\times 6!\times {}^7{C}_2$
4. $2(7!\times 8!)$

Q. The kindergarten teacher has $25$ kids in her class. She takes $5$ of them at a time, to zoological garden as often as she can, without taking the same $5$ kids more than once. Then the number of visits, the teacher makes to the garden exceeds that of a kid by

1. ${}^{25}{C}_5-{}^{24}{C}_4$
2. ${}^{24}{C}_5$
3. ${}^{25}{C}_5-{}^{24}{C}_5$
4. ${}^{24}{C}_4$