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Revision Notes for Class 11 Mathematics Chapter 7 Permutations and Combinations
Class 11 Mathematics students should refer to the following concepts and notes for Chapter 7 Permutations and Combinations in Class 11. These exam notes for Class 11 Mathematics will be very useful for upcoming class tests and examinations and help you to score good marks
Chapter 7 Permutations and Combinations Notes Class 11 Mathematics
Class XI: Math
Chapter 7: permutation and Combination
Chapter Notes
Key Concepts
1. Fundamental principle of counting: These are two fundamental principles of counting as follows:
1) Multiplication Principle
2) Addition Principle
2.Multiplication Principle: If an event can occur in M different ways, following which another event can occur in N different ways, then the total number of occurrence of the events in the given order is M x N. This principle can be extended to any number of finite events. Keyword here is “And”
3. Addition Principle: If there are two jobs such that they can be performed independently in M and N ways respectively, then either of the two jobs can be performed in (M + N) ways. This principle can be extended to any number of finite events. Keyword here is “OR”
4. The notation n! represents the product of first n natural numbers. n!=1.2.3.4…….n
5. A permutation is an arrangement in a definite order of a number of objects taken some or all at a time. In permutations order is important.
6. The number of permutation of n different objects taken r at a time, where 0< r ≤ n and the objects do not repeat is n(n – 1)(n – 2). . . (n – r + 1) which is denoted by nPr
7. The number of permutation of n different objects taken r at a time, where repetition is allowed is nr.
8. The number of permutation of n objects, where p, objects are of one kind and rest are all different is given by n!/p!.
9.The number of permutation of n objects, where p1, objects are of one kind, p2, are of second kind, … pk, are of kth kind and the rest, if any are of different kind is n!/p1!p2!....pk!
10. Keyword of permutations is “arrangement”
11. The number of combinations or selection of r different objects out of n given different objects is nCr which is given by ncr=n!/r!(n-r)!0 
12 Number of combinations of n different things taken nothing at all is considered to be 1
13. Counting combinations is merely counting the number of ways in which some or all objects at a time are selected.
14. Keyword of combinations is “selection”.
15. Selecting r objects out of n objects is same as rejecting (n – r) objects so nCn-r= nCr
16.Order is not important in combinations.
Advanced Questions and Answers for NCERT Class 11 Permutations
1. A teacher has to select either a boy or a girl from a class of 12 boys and 15 girls for coordinating a school function. In how many ways can she do it?
Answer : 27
2. There are 5 routes from A to B and 3 routes from place B to place C. Find how many different routes are there from A to C via B.
Answer : 15
3. If there are 20 buses plying between places A and B, in how many ways can a round trip from A be made if the return journey was made on:
(i) the same bus (ii) a different bus
Answer : (i) 20 (ii) 380
4. There are 4 multiple choice questions in an examination. How many sequences of answers are possible, if each question has 2 choices?
Answer : 16
5. A coin is tossed three times and the outcomes are recorded.
(i) How many possible outcomes are there?
(ii) How many possible outcomes are there if the coin is tossed n times?
Answer :(i) 8 (ii) 2n
6. Find the number of odd positive three digit integers.
Answer : 450
7. How many odd numbers less than 1000 can be formed using the digits 0, 1, 4 and 7 if repetition of digits is allowed?
Answer :32
8. Find the number of different signals that can be made by arranging at least 3 flags in order on a vertical pole, if 6 different flags are available.
Answer :1920
9. How many three digit numbers are there such that at least one of their digit is 7?
Answer : 252
10. In how many ways can five people be seated in a car with two people in the front seat and three in the rear, if two particular persons out of the five cannot drive?
Answer :72
11. In how many ways can 5 different balls be distributed among three boxes.
Answer : 243
12. How many A.P.’s with 10 terms are there whose first term belongs to the set {1, 2, 3} and common difference belongs to the set {1, 2, 3, 4, 5}.
Answer : 15
13. How many non-zero numbers can be formed using the digits 0, 1, 2, 3, 4 and 5 if repetition of digits is not allowed?
Answer : 1630
14. A class consists of 40 girls and 60 boys. In how many ways can a president, vice president, treasure and secretary be chosen if the treasure must be a girl, the secretary must be a boy and a student may not hold more than one office?
Answer : 22814400
15. Find the total number of ways in which n distinct objects can be put into two different boxes so that no box remains empty.
Answer : 2n–2
16. A team consisting of 7 boys and 3 girls plays singles matches against another team consisting of 5 boys and 5 girls. How many matches can be scheduled between the two teams if a boy plays against a girl and a girl plays against a boy?
Answer : 50
17. A movie theatre has 3 entrances and 4 exits, in how many ways can a man enter and exit from the theatre ?
Answer : 12
18. There are 3 nominations for the post of president, 4 for the post of vice-president and 5 for the secretary.
(i) In how many ways can candidates be selected for each of these posts ?
(ii) In how many ways can any one of these posts be filled ?
Answer :(i) 60 (ii) 12
19. (a) Find the number of possible outcomes of tossing a coin four times.
(b) (i) A class consists of 27 boys and 14 girls. In how many ways can one boy and one girl be selected to represent the class at a function ?
(ii) From a committee of 8 persons, in how many ways can we choose a chairman and a vice-chairman assuming that one person cannot hold more than one position.
Answer : (a) 16 (b)(i)378(ii)56
20. Number 1, 2 and 3 are written on three cards. How many two digit numbers can be formed by placing two cards side by side ?
Answer : 6
21. A person wants to go to another city by bus and return by train. He has a choice of 5 different buses and 4 trains to return. In how many ways can be perform his journey?
Answer : 20
22. Eight children are standing in a queue.
(i) In how many ways can the queue be formed ?
(ii) How many arrangements are possible if the tallest child stands at the end of the queue ?
Answer : (i) 40320 (ii) 5040
23. In how many ways can a student answer a set of ten true/false type question ?
Answer :1024
24. How many numbers are there between 100 and 1000 in which all the digits are distinct ?
Answer :648
25. There are seven flags of different colours. A signal is generated using two flags How many different signals can be generated ?
Answer : 42
26. How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5, if
(i) repetition of digits is allowed.
(ii) repetition of digits is not allowed.
Answer : (i) 125 (ii) 60
27. How many numbers can be formed from the digits 1, 2, 3 and 9, if repetition of digits is not allowed?
Answer :64
28. There are 6 multiple choice questions in an examination. How many sequence of answers are possible, if the first three questions have 4 choices each and the next three have 5 each ?
Answer : 8000
29. How many three digit numbers with distinct digits are there whose all the digits are odd?
Answer :60
30. The first ten English alphabets are written on slips of paper and placed in a box. Three of the slips are drawn and placed in order. How many arrangements are possible?
Answer : 720
31. How many numbers of four digits can be formed with the digits 1, 2, 3, 4 and 5? (if repetition of digits is not allowed).
Answer : 120
32. How many numbers between 400 and 1000 can be made with the digits 2, 3, 4, 5, 6 and 0 when repetition of digits is not allowed?
Answer : 60
33. How many numbers greater than 40000 can be formed using the digits 1, 2, 3, 4 and 5 if each digit is used only once in each number?
Answer : 48
34. How many different 4-digit numbers can be formed from the digits 2, 3, 4 and 6 if each digit is used only once in a number? Further, how many of these numbers (i) end in a 4? (ii) end in a 3? (iii) end in a 3 or 6?
Answer : 24 (i) 6 (ii) 6 (iii) 12
35. Find the number of numbers between 300 and 3000 which can be formed with the digits 0, 1, 2, 3, 4 and 5, no digit being repeated in any number.
Answer : 180
36. How many odd numbers greater than 80000 can be formed using the digits 2, 3, 4, 5 and 8 if each digit is used only once in each number?
Answer : 12
37. How many even numbers of four digits can be formed with the digits 0, 1, 2, 3, 4, 5 and 6; no digit being used more than once?
Answer : 420
38. How many even numbers are there with three digits such that if 5 is one of the digits in a number then 7 is the next digit in that number.
Answer : 365
39. How many numbers of six digit can be formed from the digits 0, 1, 3, 5, 7 and 9 when no digit is repeated? How many of them are divisible by 10?
Answer : 600, 120
40. How many positive numbers can be formed by using any number of the digits 0, 1, 2, 3 and 4; no digit being repeated in any number?
Answer : 260
41. How many different numbers can be formed by using all the digits 1, 2, 3, 4, 3, 2, 1, so that odd digits always occupy odd places?
Answer : 18
42. How many numbers greater than a million can be formed with the digit 2, 3, 0, 3, 4, 2 and 3 if repetition of digit is not allowed?
Answer : 360
43. How many natural numbers not exceeding 4321 can be formed with the digits 1, 2, 3 and 4, if the digits can repeat?
Answer : 10 × 95
44. How many number of four digits can be formed with the digits 1, 2, 3, 4? Find the sum of those numbers?
Answer : 313
Advanced Questions and Answers for NCERT Class 11 Combinations
1. Find the value of (2n)! /n! :
Answer : {1.3.5.....(2n – 1)} 2n
2. If n! /2!(n - 2)! and n! /4!(n - 4)! are in the ratio 2 : 1, then find the value of n
Answer : 5
3. Find the exponent of 2 in 50!?
Answer : 47
4. Find the number of zeroes in 100!
Answer :24
5. There are 15 gates to enter a city from north and 10 gates to enter the city from east. In how many ways a person can enter the city?
Answer : 25
6. There are 15 students in a class in which 10 are boys and 5 are girls. The class teacher selects either a boy or a girls for monitor of the class. In how many ways the class teacher can make this selection?
Answer :15
7. There are 3 routes to travel from City A to City B and 4 routes to travel from City B to City C and 7 routes from C to D. In how many different ways (routes) a man can travel from City A to City D via City B and City C.
Answer : 84
8. A city has 12 gates, In how many ways can a person enter the city through one gate and come out through a different gate?
Answer : 132
9. How many (a) 5-digit (b) 3-digit numbers can be formed using 1, 2, 3, 7, 9 without any repetition of digits?
Answer :(a) 120 (b) 60
10. How many 3-letter words can be formed using a, b, c, d, e if :
(a) repetition is not allowed (b) repetition is allowed?
Answer : (a) 60 (b) 125
11. How many four-digit distinct numbers can be formed using the digits 0, 1, 2, 3, 4, 5 ?
Answer : 300
12. In how many ways can six persons be arranged in a row?
Answer : 720
13. How many 5-digit odd numbers can be formed using digits 0, 1, 2, 3, 4, 5 without repeating digits?
Answer : 288
14. How many 5-digit numbers divisible by 2 can be formed using digits 0, 1, 2, 3, 4, 5 without repetition of digits.
Answer : 312
15. How many 5-digits numbers divisible by 4 can be formed using digits 0, 1, 2, 3, 4, 5, without repetition of digits.
Answer : 144
16. How many six-digit numbers divisible by 25 can be formed using digits 0, 1, 2, 3, 4, 5?
Answer :42
17. Find number of different words which can be formed using all the letters of the word ‘HISTORY’.
Answer : 5040
18. In how many ways 5 different red balls, 3 different black balls and 2 different white balls can be arranged along a row?
Answer : 10 !
19. In how many ways can the letters of the word ‘DELHI’ be arranged so that the vowels occupy only even places ?
Answer : 12
20. (a) How many words can be made by using letters of the word COMBINE all at a time?
(b) How many of these words began and end with a vowel?
(c) In how many of these words do the vowels and the consonants occupy the same relative positions as in COMBINE?
Answer : (a) 5040 (b) 720 (c) 144
21. (a) How many words can be formed using letters of the word EQUATION taken all at a time?
(b) How many of these begin with E and end with N?
(c) How many of these end and begin with a consonants?
(d) In how many of these vowels occupy the first, third, fourth, sixth and seventh positions?
Answer : (a) 40320 (b) 720 (c) 4320 (d) 720
22. Find number of different 4 letter words can be formed using the letters of the word ‘HISTORY’.
Answer : 840
23. In how many ways 5 different red balls, 3 different black balls and 2 different white balls can be placed in 3 different boxes such that each box contains only 1 ball.
Answer : 720
24. How many nine-letter words can be formed by using the letters of the words (a) E Q U A T I O N S (b) A L L A H A B A D ?
Answer : (a) 362880 (b) 7560
25. In how many ways can 5 letters be posted in 4 letter boxes?
Answer : 45
26. Five person entered the lift cabin on the ground floor of an 8-floor house. Suppose each of them can leave the cabin independently at any floor beginning with the first. Find the total number of ways in which each of the five persons can leave the cabin
(i) at any one of the 7 floors (ii) at different floors
Answer : (i) 75 (ii) 2520
27. There are 6 single choice questions in an examination. How many sequence of answers are possible, if the first three questions have 4 choices each and the next three have 5 each?
Answer :2520
28. Three tourists want to stay in five different hotels. In how many ways can they do so if :
(a) each hotel can not accommodate more than one tourist?
(b) each hotel can accommodate any number of tourist?
Answer : (a) 60 (b) 125
Please click the link below to download pdf file for CBSE Class 11 Mathematics - Permutation and Combination Concepts.
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CBSE Class 11 Mathematics Chapter 7 Permutations and Combinations Notes
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