Rs aggarwal Maths CBSE Class 11 Chapter 9 Exercise 9B
There are 18 points in a plane of which 5 are collinear. How many straight lines can be formed by joining them?
Answer : A line is formed by joining two points. If the total number of points is 18, the total number of lines would be = 18C2 But 5 points are collinear, so the lines made by these points are the...
Find the number of ways in which a committee of 2 teachers and 3 students can be formed out of 10 teachers and 20 students. In how many of these committees
(i)a particular teacher is included?
(ii)a particular student is included?
(iii)a particular student is excluded?
Answer : Since a committee is to be formed of 2 teachers and 3 students When a particular teacher is included No. of ways in which committee can be formed = 9C1 20C3 = 9720 ways A particular...
How many different products can be obtained by multiplying two or more of the numbers 3, 5, 7, 11 without repetition?
Answer : The given no. is 3,5,7,11. The no. of different products when two or more is taking= the no. of ways of taking the product of two no.+ the no. of ways of taking the product of three no. +...
How many different selections of 4 books can be made from 10 different books, if
(i)there is no restriction?
(ii)two particular books are always selected?
(iii)two particular books are never selected?
Answer : Since there are 10 different books out of which 4 is to be selected . When there is no restriction No. of ways in which 4 books be selected = 10C4 = 210 ways Two particular books are always...
How many triangles can be formed in a decagon?
Answer : Total number of sides in a decagon = 10 We know that number of vertices in triangle = 3 So, out of 10 vertices we have to choose 3 vertices. Therefore, Total number of triangles in a...
How many triangles can be obtained by joining 12 points, four of which are collinear?
Answer : Total number of points on plane = 12 Triangles can be formed from these points = 12C3 = 220 But 4 points are colinear, the number of triangles can be formed from these points = 4C3 = 4 We...
Find the number of diagonals of
(i)a hexagon,
(ii)a decagon,
(iii)a polygon of 18 sides
Answer : For a diagonal to be formed, 2 vertices are required. Thus in a polygon, there are 10 sides. And no. of lines can be formed are nC2, but in nC2 the sides are also included. N of them is...
Find the number of 5-card combinations out of a deck of 52 cards if a least one of the five cards has to be king.
Answer : Since there are 52 cards in a deck out of which 4 are king and others are non- kings. So, the no. of ways are as follows: 1 king and 4 non-king 2 king and 3 non-king 3 king and 2 non-king 4...
From a class of 14 boys and 10 girls, 10 students are to be chosen for a competition, at least including 4 boys and 4 girls. The 2 girls who won the prizes last year should be included. In how many ways can the selection be made?
Answer : 2 girls who won the prize last year are surely to be taken. So, we have to make a selection of 8 students out of 14 boys and 8 girls, choosing at least 4 boys and at least 2 girls. Thus, we...
A committee of 5 is to be formed out of 6 gents and 4 ladies. In how many ways can this be done, when (i) at least 2 ladies are included? (ii) at most 2 ladies are included?
Answer : Since the committee of 5 is to be formed from 6 gents and 4 ladies. (i) Forming a committee with at least 2 ladies Here the possibilities are 2 ladies and 3 gents 3 ladies and 2 gents 4...
A committee of three 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?
Answer : Total number of persons = 2 + 3 = 5 Now, committee consist of 3 persons. Therefore, total number of ways = 5C3 = 5 × 2 = 10 Now, When 1 man is selected, total ways = 2C1 When 2 women are...
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
(i)exactly 3 girls?
(ii)at least 3 girls?
(iii)at most 3 girls?
Answer : A committee of 7 has to be formed from 9 boys and 4 girls. Exactly 3 girls: If there are exactly 3 girls in the committee, then there must be 4 boys, and the ways in which they can be...
Out of 6 teachers and 8 students, a committee of 11 is being formed. In how many ways can this be done, if the committee contains
(i)exactly 4 teachers?
(ii)at least 4 teachers?
Answer : Since the committee of 11 is to be formed from 6 teachers and 8 students. Forming a committee with exactly 4 teachers Choosing 4 teachers out of 6 in 6C4 Remaining 7 from 8 students in 8C7...
In an examination, a candidate is required to answer 7 questions out of 12, which are divided into two groups, each containing 6 questions. One cannot attempt more than 5 questions from either group. In how many ways can he choose these questions?
Answer : There are total 13 questions out of which 10 is to be answered .The student can answer in the following ways: ⇒ 3 questions from part A and 4 from part B ⇒ 4 questions from part A and 3...
In an examination, 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 these questions be chosen?
Answer : There are total 13 questions out of which 10 is to be answered .The student can answer in the following ways: ⇒ 6 questions from part A and 4 from part B ⇒ 5 questions from part A and 5...
In an examination, a student has to answer 4 questions out of 5. Questions 1 and 2 are compulsory. Find the number of ways in which the student can make a choice.
Answer : A student has to answer 4 questions out of 5 in which he is compelled to do the 1 and 2 questions compulsory. So he has to attempt 2 questions from 3 of his choice. Choosing 2 questions...
A question paper has two parts, part A and part B, each containing 10 questions. If the student has to choose 8 from part A and 5 from part B, in how many ways can he choose the questions?
Answer : The question paper has two sets each containing 10 questions. So the student has to choose 8 from part A and 5 from part B. ⇒ choosing 8 questions from 10 of part A in 10C8 ⇒ choosing 5...
In how many ways can a cricket team be selected from a group of 25 players containing 10 batsmen, 8 bowlers, 5 all-rounders and 2 wicketkeepers, assuming that the team of 11 players requires 5 batsmen, 3 all-rounders, 2 bowlers and 1 wicketkeeper?
Answer : A team of 11 players is to be made from 25 players. ⇒ Selecting 5 batsmen from 10 in 10C5 ways. ⇒ Selecting 3 all-rounders from 5 in 5C3 ways. ⇒ Selecting 2 bowlers from 8 in 8C2 ways. ⇒...
A cricket team of 11 players is to be selected from 16 players including 5 bowlers and 2 wicketkeepers. In how many ways can a team be selected so as to consist of exactly 3 bowlers and 1 wicketkeeper?
Answer : There is a cricket team of 11 players is to be selected from 16 players, which must include 3 bowlers and a wicketkeeper. ⇒ There will be a team of 7 batsmen, 1 wicketkeeper and 3 bowlers....
From 4 officers and 8 clerks, in how many ways can 6 be chosen
(i) to include exactly one officer,
(ii) to include at least one officer?
Answer : The team of 6 has to be chosen from 4 officers and 8 clerks. There are some restrictions which are To include exactly one officer In this case , One officer will be chosen from 4 in 4C1...
A sports team of 11 students is to be constituted, choosing at least 5 from class XI and at least 5 from class XII. If there are 20 students in each of these classes, in how many ways can the team be constituted?
Answer : There are 20 students in each classes and there is need of at least 5 students in each class to form a team of team of 11. Now, There are two ways in which the selection can be possible...
In How many ways can a student chose 5 courses out of 9 courses if 2 specific courses are compulsory for every student?
Answer : Since every student needs to choose 5 courses out of which 2 are compulsory. So, the student needs to choose 3 subjects out of 7. No. of ways for choosing 3 subjects out of 7 is 7C3 = 35...
How many different boat parties of 8 consisting of 5 boys and 3 girls can be made from 20 boys and 10 girls.
Answer : Number of ways of choosing 5 boys out of 20 boys = 20C5 = 15 × 8 = 120 Total number of ways = 120 × 15,504 = 1,860,480 OR Total number of ways = 20C5 × 10C3
Find the number of ways of selecting 9 balls from 6 red balls, 5 while balls and 4 blue balls if each selection consists of 3 balls of each colour.
Answer : Total number of red balls = 6 Total number of white balls = 5 Total number of blue balls = 4 No. of ways of selecting 3 balls which is red = 6C3 No. of ways of selecting 3 balls which is...
A bag contains 5 black and 6 red balls. Find the number of ways in which 2 black and 3 red balls can be selected.
Answer : There are 5 black and 6 red balls. So, The number of ways of selecting 2 black balls from 5 black balls is 5C2, and number of ways of selecting 3 red balls from 6 red balls is 6C3. Thus...
In how many ways can 5 sportsmen be selected from a group of 10?
Answer : As there are 10 sportsmen out of which 5 are to be selected. 5 sportsmen can be selected out of 10 in 10C5 ways.