Exercise 15.2
Page no 15.23
Type 2
Question 1
In how many ways can a committee be selected from 15 persons if the committee is to have
(i) 3 members
(ii) 13 members
Question 2
How many different teams of 7 players can be chosen from 10 players?
Question 3
Sudha wants to choose any 9 stamps from a set of 11 different stamps. How many different selections can she make ?
Page no 15.24
Question 4
How many lines can be drawn through 21 points on a circle ?
Question 5
Seven points lie on a circle. How many chords can be drawn by joining there points?
Question 6
How many selection of 4 books can be made from 8 different books?
Question 7
In how many ways can a student choose a Programme of 5 courses if 9 courses are available and 2 courses are compulsory for every student?
Question 8
A man has 7 friends and he wants to invite 3 of them at a party. Find how many parties to each of 3 different friends he can give and how many times any particular friend will attend the parties ?
Question 9
A man has 7 friends and he wants to invite 3 of then at a party . find how many parties to each of 3 different friends he can give and how many times any particular friend will attend the parties?
Question 10
Prove that the number of combinations of $n$ things taken $r$ at a time in which $p$ particular things always occur is $n-p C_{r-} p$.
Question 11
A delegation of 6 members is to be sent abroad out of 12 members. In haw many ways can the selection be made so that
(i) A particular member is included?
(ii) A particular member is excluded?
Question 12
There are 6 students $A, B, C, D, E, F$.
(i) In how many ways can they be seated in a line so that $C$ and $D$ do not sy together ?
(ii) In how many ways can a committee of 4 be formed so as to always include $C$ ?
(iii) In how many ways can a committee of 4 be formed so as to always include $C$ but exclude $E$ ?
Question 13
(i) There are $n$ stations on a railway line. The number of kinds of tackets printed (no return tickets) is 105 . Find the number of stations.
(ii) Twelve persons meet in a room and each shakes hands with all others. Find the number of hand-shakes.
Question 14
(i) Determine the number of 5 cards combinations out of a pack of 52 playing cards if there is exactly one ace in each combination.
(ii) Determine the number of 5 cards combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.
Question 15
There are 15 points in a plane, no three of which are collinear. Find the number of triangles formed by joining them.
Type 3
Question 16
There are 10 points in a plane of which 4 are collinear. No three of the remaining 6 points are collinear. How many different straight lines can be drawn by joining them ?
Question 17
There are 15 points in a plane, no three of which are in the same straight line with the exception of 6 , which are all in the same straight line. Find the number of (t) straight lines formed (ii) number of triangles formed by joining thee? points.
Question 18
There are 10 points in a plane out of which 5 are collinear. Find the number of quadrilaterals formed having vertices at these points.
Page no 15.25
Question 19
The sides $A B, B C, C A$ of a triangle $A B C$ have 3,4 and 5 interior points respectively on them. Find the number of triangles that can be constructed using given interior points as vertices.
Question 20
(i) In how many ways can a team of 11 be chosen from 14 football players if two of them can be only goalkeepers ?
(ii) In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers.
Question 21
To fill 12 vacancies there are 25 candidates of which 5 are from scheduled castes . If 3 of the vacancies are reserved for scheduled caste candidates while the rest are open to all. Find the number of ways in which the selections can be made.
Question 22
(i) A committee consisting of 2 men and 2 women is to be chosen from 5 men and 6 women. In how many ways can this be done?
(ii) A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how many ways can this be done? How many of these committees would consist of 1 man and 2 women ?
Question 23
(i) There are 6 boys and 3 girls in a class. An entertainment committee of 6 persons is to be selected such that there are 4 boys and 2 girls in the committee. In how many ways can the committee be selected ?
(ii) How many different committees each consisting of 3 girls and 2 boys can be chosen from 7 girls and 5 boys ?
Question 24
24. What is the number of ways of choosing 4 cards form a pack of 52 playing cards ? In how many of these :
$[H O T S]$
(i) Four cards are of the same suit ?
(ii) Four cards belong to four different suits ?
(iii) Are face cards ?
(iv) Two are red cards and two are black cards ?
(v) Cards are of the same Colour?
Question 25
A bookshelf contains 7 different mathematics textbooks and 5 different physics textbooks. How many groups of 3 mathematics and 3 physics textbooks can be selected?
Question 26
Find the number of ways of selecting 9 balls from 6 red balls. 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.
Question 27
In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls ?
Question 28
A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected from lot.
Question 29
A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of at most 3 girls?
Question 30
At an election three wards of a town are convassed by 4,5 and 8 men respectively. If there are 20 volunteers. In how many ways can they be allotted to different wards ?
Question 31
Out of 7 men and 4 ladies a committee of 5 is to be formed. In how many Ways can this be done so as to include at least 3 ladies ?
Page no 15.26
Question 32
A candidate is required to answer six out of ten questions . Which are divided into two groups each containing five question and he is not permitted to attempt more than 4 from any group . In how many ways can he make up his choice ?
Question 34
In an examination, a question paper consists of 12 questions divided into two parts $I$ and 11, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions ?
Question 35
There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed. Find in how many ways these committee can be formed if
(i) a particular professor is included?
(ii) a particular professor is excluded ?
Question 36
From 6 boys and 7 girls a committee of 5 is to be formed so as to include at least one girl. Find the number of ways in which this can be done.
Question 37
From 6 gentlemen and 4 ladies, a committee of 5 is to be formed. In how many ways can this be done if
(i) there is no restriction?
(ii) the committee is to include at least one lady?
Question 38
From 8 gentlemen and 4 ladies, a committee of 5 is to be formed. In how many ways can this be done so as to include at least one lady ?
Question 39
In a group of 15 boys, there are 6 hockey players. In how many ways can 12 boys be selected so as to include at least 4 hockey players ?
Question 40
From 7 gentlemen and 4 ladies a committee of 5 is to be formed. In how many ways can this be done so as to include at least one lady ?
Question 41
From 7 gentlemen and 5 ladies, a boat party of 5 is to be formed. In how many ways can this be done so as to include at least one lady ?
Question 42
A committee of 6 is to be formed out of 4 boys and 6 girls. In how many Ways can it be done so that the girls may not be outnumbered?
Question 43
A person has 12 friends of whom 8 are relatives. in how many ways can he invite 7 friends such that at least 5 of them may be relatives?
Question 44
A candidate is required to answer 7 questions out of 12 questions which are divided into two groups , each containing 6 questions he is not permitted to attempt more than 5 from either group In how may different ways can he choose the seven question ?
Question 45
Each of two parallel lines has a number of distinct points marked on then on one line there are 2 points s $P$ and $Q$ and on the other there are 8 points
Question 46
There are 7 men and 3 ladies contesting for two vacancies an elector can vote for any number of candidates not exceeding the number of vacancies . In how many ways can he vote?
Question 47
A party of 6 is to be formed from 10 boys and 7 girls so as to include 3 boys and 3 girls . In how many different ways can the party be formed if two particular girls refuse to join the same party ?
Page no 15.27
Question 48
In an examination, the question paper contains three different sections $A, B$ and $C$ containing 4,5 and 6 questions respectively. In how many ways, a candidate can make a selection of 7 questions, selecting at least two questions from each section.
Type 4
Question 49
From 5 apples, 4 oranges and 3 mangoes, how many selections of fruits can be made ?
Question 50
Find the total number of selections of at least one red ball from 4 red and 3 green balls if the balls of the same colour are different.
Question 51
Find the number of different sums that can be formed with one rupee, one half rupee and one quarter rupee coins.
Question 52
There are 5 questions in a question paper. In how many ways can a boy solve one or more questions ?
Question 53
In an election for 3 seats there are 6 candidates. A voter cannot vote for more than 3 candidates. In how many ways can he vote ?
Question 54
In an election the number of candidates is one more than the number of members to be elected. If a voter can vote in 30 different ways, find the number of candidates (A voter has to vote for at least one candidates).
Type 5
Question 55
In how many ways 12 different books can be distributed equally among 4 persons ?
[HOTS]
Question 56
In how many ways 12 different books can be distributed equally among 3 persons ?
Question 57
In how many ways can a pack of 52 playing cards be divided in 4 sets three of them having 17 cards each and the fourth just one card?
(Image to be added)
Question 58
In how many ways can 7 cross marks ' $x$ ' be placed in the given figure so that each row has at least one cross mark?
Question 59
Five crosses are to be put into eleven square blocks written in the form of an $E$ as shown in figure so that every row has a cross. In how many different ways can it be done ?
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