Please refer to Permutations and Combinations Class 11 Mathematics Exam Questions provided below. These questions and answers for Class 11 Mathematics have been designed based on the past trend of questions and important topics in your class 11 Mathematics books. You should go through all Class 11 Mathematics Important Questions provided by our teachers which will help you to get more marks in upcoming exams.

## Class 11 Mathematics Exam Questions Permutations and Combinations

Class 11 Mathematics students should read and understand the important questions and answers provided below for Permutations and Combinations which will help them to understand all important and difficult topics.

**Question. 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 :** n = 10

**Question. 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

**Question. A movie theatre has 3 entrances and 4 exits, in how many ways can a man enter and exit from the theatre ?****Answer :** 12

**Question. 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

**Question. 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) 2^{n}

**Question. Find the number of odd positive three digit integers.****Answer :** 450

**Question. 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

**Question. Four books, one each in Chemistry, Physics, Biology and Mathematics, are to be arranged on a shelf. In how many ways can this be done ?****Answer :** 24

**Question. How many different signals can be generated from 6 flags of different colours if each signal makes use of all the flags at a time, placed one below the other ?****Answer : **720

**Question. Six candidates arc called for interview to fill four posts in an office. Assuming that each candidate is fit for each post, determine the number of ways in which**

(i) first and second posts can be filled,

(ii) first three posts can be filled,

(iii) if all the four posts can be filled.**Answer : **(i) 30 (ii) 120 (iii) 360

**Question. In how many ways can 4 red, 3 yellow and 2 green discs be arranged in a row if the discs of the same colour are indistinguishable ?****Answer : **1260

**Question. Find the total number of signals that can be made by five flags of different colours when any number of them may be used.****Answer :** 325

**Question. Serial numbers for an item produced in a factory are to be made using two letters followed by four digits (0 to 9). If the letters are to be taken from the first six letters of English alphabet without repetition and the digits are also not repeated in a serial number, how many serial numbers are possible?****Answer :** 151200

**Question. Find the sum of all the 4 digit numbers that can be formed with the digits 3, 2, 3, 4.****Answer : **39996

**Question. In the given figure, we see that it has 4 horizontal blocks (or paths) and 3 vertical blocks (or paths). This is known as 4 × 3 grid. Seema wishes to go from A to B, but the instruction is that she must go only on the right and only up, but not necessarily in that order. How many possible paths does she have at her disposal ?**

**Answer :** 35

**Question. Find the number of arrangements of the letters of the word ‘Delhi’ if ‘e’ always come before ‘i’.****Answer :** 5!/2! = 60

**Question. Find the number of different 8-letter arrangements that can be made from the letters of the word DAUGHTER so that**

(i) all vowels occur together

(ii) all vowels do not occur together.**Answer :** (i) 4320 (ii) 36000

**Question. Find the number of different arrangements (permutations) of the letters of the word ‘B ANN AN A’.****(ii) Find the number of permutations of the letters of the word ALLAHABAD.****Answer : **(i) 140 (ii) 7560

**Question. In how many ways can the letters of the word ‘INTERMEDIATE’ be arranged among themselves so that no two vowels may occupy consecutive places?****Answer :** 151200

**Question. In how many ways can the letters of the word ‘PERMUTATIONS’ be arranged if the**

(i) words start with P and end with S ?

(ii) vowels are all together

(iii) there are always 4 letters between P and S**Answer :** (i) 1814400 (ii) 2419200 (iii) 25401600

**Question. How many numbers greater than 1000 but not greater than 4000 can be formed with the digits 0, 1, 2, 3, 4, repetition of digits being allowed.****Answer : **375

**Question. 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

**Question. 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

**Question. How many numbers can be formed from the digits 1, 2, 3 and 9, if repetition of digits is not allowed?****Answer :** 64

**Question. 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

**Question. 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

**Question. 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

**Question. How many number of four digits can be formed with the digits 1, 2, 3, 4? Find the sum of those numbers?****Answer :** 24; 66660

**Question. In how many ways 3 prize can be given away to 7 boys when each boy is eligible for any of the prizes.****Answer : **343

**Question. In how many different ways the following 5 prizes be distributed among 10 students. First and sec- ond in Mathematics; first and second in Physics and first in Hindi.****Answer : **81000

**Question. There are stalls for 12 animals in a ship. In how many ways the shipload can be made if there are cows, calves and horses to be transported, animals of each kind being not less than 12 ?****Answer :** 3^{12}

**Question. A telegraph has 5 arms and each arm is capable of 4 distinct positions, including the position of rest. What is the total number of signals that can be made.****Answer : **1023

**Question. In how many ways can three prizes be given away to 5 students when each student is eligible for any of the prizes ?****Answer :** 125

**Question. In how many ways can n things be given to p persons, when each person can get any number of things (n > p).****Answer :** p^{n}

**Question. (Secret Lock Problem): A letter lock consists of three rings each marked with 10 different letters. In how many ways is it possible to make an unsuccessful attempt to open the lock.****Answer :** 999

**Question. Find the number of ways in which one can post 4 letters in 6 letter boxes.****Answer :** 1296

**Question. In how many ways can 4 different balls be distributed among 5 boxes, when**

(i) no box has more than one ball.

(ii) a box can have any number of balls.

(iii) no box contains all the balls.**Answer :** (i) 120 (ii) 625 (iii) 620

**Question. The lock of a safe has a dial with holes, say ten, in which the numbers 0, 1, 2, …, 9 are inscribed in each hole. The lock can be opened only when a specific ‘code number’ say of six digits, is dialled. Suppose the ‘code number’ is 249916, it means that the lock can be opened when we first dial 2, then 4 and so on. Find the maximum number of trials which do not result in opening the lock.****Answer : **999999

**Question. How many words can be formed from the letters of the word ‘CIRCUMFERENCE’ taken all together?****Answer : **13!/3!3!2! = 86486400

**Question. In how many ways can three jobs I, II and III be assigned to three persons A, B and C. If one person in assigned only one job and all are capable of doing each job? Which assignment of job will take the least time to complete the jobs, if time taken (in hours) by an individual on each job is as follows?**

**Answer :** 11½ hours

**Question. Find the number of permutations of the letters of the word ‘INDEPENDENCE’, how many of these arrangements**

(i) do the words start with P ?

(ii) do all the vowels always occur together ?

(iii) do the vowels never occur together ?

(iv) do the words begin with I and end in P.**Answer : **12!/3!2!4! = 1663200 (ii) 138600 (iii) 16800 (iv) 146400

**Question. How many words can be formed using all letters of the word, EQUATION, so that**

(i) each letter occurs exactly once ?

(ii) vowels and consonants occur together ?**Answer :** (i) 40320 (ii) 1440

**Question. In how many ways can 7 I. A and 5 I.Sc. students be seated in a row so that no two of the I.Sc. students may sit together?****Answer :** ^{8}P_{5} × 7 !

**Question. In a class of 10 students, there are 3 girls. In how many different ways can they be arranged in a row such that no two of the three girls are consecutive?****Answer :** 42(8!)

**Question. The letters of the word TUESDAY are arranged in a line, each arrangement ending with letter S. How many different arrangements are possible ? How many of them start with letter D ?****Answer :** 720; 120

**Question. How many different words can be formed with the letters of the word ‘MATHEMATICS’ ? In how many of them, the vowels are together and consonants are together?****Answer : **11!2!7!4! / 2!2!2!2!2!2!

**Question. In how many ways can the letters of the word ‘BANARAS’ be arranged so that the letters N and S are never together?****Answer :** 600

**Question. A library has two books each having three copies and three other books each having two copies. In how many ways can all these books be arranged in a shelf so that copies of the same book are not separated.****Answer :** 120

**Question. Three married couples are to be seated in a row having six seats in a cinema hall. If spouses are to be seated next to each other, in how many ways can they be seated? Find also the number of ways of their seating if all the ladies sit together.****Answer :** 48; 144

**Question. In how many ways can the letters of the word ‘PLANTAIN’ be arranged so that the two ‘A’ do not come together?****Answer : **7560

**Question. If repetition of digits is not allowed, how many different numbers of 6 digits each can be formed with the digits 4, 5, 6, 7, 8, 9 ? How many of them are not divisible by 5.****Answer :** 720;600

**Question. How many different numbers can be formed with the digits 1, 3, 5, 7 and 9; when taken all at a time and what is their sum ?****Answer :** 120, 6666600

**Question. Find the sum of all the 4 digit numbers that can be formed with the digits 0, 2, 3, 5.****Answer : **64440

**Question. How many even numbers of 5 digits without repetition can be formed with the digits 1, 2, 3, 4 and 5.****Answer :** 48

**Question. How many four digit numbers are there, with no digit repeated?****Answer :** 4536

**Question. Each of the six squares in the strip shown in the figure given below is to be coloured with any one of ten different colours so that no two adjacent squares have the same colour. Find the number of ways of colouring the strip.**

**Answer :** 10 × 9^{5}

**Question. How many natural numbers not exceeding 4321 can be formed with the digits 1, 2, 3 and 4, if the digits can repeat?****Answer :** 313

**Question. How many numbers of 5 digits can be formed with the digit 0, 1, 2, 3, 4 no digit being repeated?****Answer :** 96

**Question. Find the number of 4-digits numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is used more than once in a number. How many of these numbers will be even?****Answer :** 120; 48

**Question. Find the total natural numbers from 1 to 1000 have none of their digits repeated?****Answer :** 738

**Question. In how many ways can a ten question multiple choice examination be answered there are four choices a, b, c and d to each question ? If no two consecutive questions are answered the same way, how many ways are there ?****Answer :** 4^{10}; 4 × 3^{9}

**Question. Find the number of numbers of four digits that can be made from the digits 0, 1, 2, 3, 4, 5 if digits can be repeated in the same number. How many of these numbers have at least one digit repeated ?****Answer :** 1080; 780

**Question. How many numbers each lying between 1000 and 10000 can be formed with the digits 0, 1, 2, 3, 4, 5; no digit being repeated?****Answer : **300

**Question. Find the number of words formed by the letters of the word ‘DELHI’ which**

(i) begin with D

(ii) end with I

(iii) the letter L being always in the middle

(iv) begin with D and end with I**Answer :** (i) 24 (ii) 24 (iii) 24 (iv) 6

**Question. How many words can be formed out of the letters of the word ‘ARTICLE’ so that the vowels occupy the even places.****Answer : **144

**Question. In how many different ways, the letters of the word ALGEBRA can be arranged in a row if**

(i) the two A’s are together ? (ii) the two A’s are not together?**Answer :** 42. (i) 720 (ii) 1800

**Question. How many words can be formed with the letters of the word ‘PATALIPUTRA’ without changing the relative order of the vowels and consonants?****Answer :** 3600

**Question. There are 5 boys and 3 girls. In how many ways can they be seatd in a row so that all the three girls do not sit together.****Answer :** 36000

**Question. In how many ways 5 boys and 3 girls be seated in a row so that no two girls are together?****Answer : **14400

**Question. In how many ways can the letters of the word ‘MUZAFFARPUR’ be rearranged? How many such words will begin with M ?****Answer :** 11! / (2!)^{4} – 1; 10! / (2!)^{4}

**Question. In how many ways can the letters of the word ASSASSINATION be arranged so that all the S’s are together ?****Answer : **151200

**Question. Given ^{2n+1}P_{n-1} : ^{2n-1}P_{n} = 3 : 5.**

**Answer :**n = 4

**Question. 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 :** r = 3

**Question. How many different numbers greater than 5000 can be formed with the digits 0, 1, 5, 9; no digit being repeated?****Answer : **12

**Question. Find the number of numbers lying between 300 and 4000 that can be formed with the digits 0, 1, 2, 3, 4, 5; no digit being repeated?****Answer :** 240

**Question. 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

**Question. There are 4 multiple choice questions in an examination. How many sequences of answers are possible, if each question has 2 choices?****Answer :** (i) 16

**Question. If repetition of digit is not allowed, how many numbers of four digit divisible by 5 can be formed with the digits 0, 4, 5, 6, 7?****Answer :** 42

**Question. Find the number of numbers of 5 digits that can be formed with the digits 0, 1, 2, 3 and 4 if repetition of digits is allowed.****Answer :** 2500

**Question. In how many ways 6 rings of different types can be had in 4 given fingers of a hand ?****Answer :** 4^{6}

**Question. (i) Find the number of numbers of six digits without repetition formed with the digit 1, 2, 3, 4, 5, 6 in which 5 always occurs in the tens place.****(ii) How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which are divisible by 10 and no digit is repeated.****Answer :** (i) 120 (ii) 120

**Question. Find the number of positive integers which can be formed by using any number of digits from 0, 1, 2, 3, 4, 5 but using each digit not more than once in each number. How many of these integers are greater than 3000 ?****Answer : **1630, 1380

**Question. A servant has to post 5 letters and there are 4 letter boxes. In how many ways can post the letters ?****Answer :** 1024

**Question. Find the number of functions that can be defined from A to B if number of distinct elements in A and B are m and n respectively.****Answer :** n^{m}

**Question. In how many ways 5 delegates can be put in 6 hotels of a city if there is no restriction ?****Answer :** 6^{5}

**Question. In how many ways can 5 different balls be distributed among three boxes.****Answer :** 243

**Question. 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

**Question. How many numbers less than 1000 and divisible by 5 can be formed in which no digit occurs more than once in the same number?****Answer : **154

**Question. Find how many numbers between 100 and 999 can be formed with digits 0, 4, 5, 6, 7, 8; no digit being used more than once. How many of them are odd?****Answer :** 100, 32

**Question. From a pool of 12 candidates, in how many ways can we select president, vice-president, secretary and a treasurer if each of the 12 candidates can hold any office?****Answer :** 11880

**Question. From a committee of 8 persons, in how many ways can we choose a chairman and a vice-chairman assuming one person cannot hold more than one position ?****Answer :** 56

**Question. In how many ways can the letters of the word ‘ARRANGE’ be arranged so that**

(i) the two R’s are never together?

(ii) the two A’s are together but not the two R’s?

(iii) neither the two A’s nor the two R’s are together?**Answer :** (i) 900 (ii) 240 (iii) 660

**Question. It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible?****Answer :** 2880

**Question. How many even numbers of three digits each can be made with the digits 1, 2, 3, 4, 6, 7 if no digit is repeated?****Answer :** 60

**Question. 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

**Question. How many 4-letter codes can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?****Answer : **5040

**Question. How many numbers of four digits can be formed with the digits 1, 2, 4, 5, 7 no digit being repeated?****Answer :** 120