CBSE Class 11 Mathematics Permutations And Combinations Worksheet Set 02

Question. In how many ways can 15 I.Sc and 12 B.Sc. candidates be arranged in a line so that no two B.Sc candidates may occupy consecutive positions?
Answer : 15! × ¹⁶P

Question. How many different words can be formed with the letters of the word ‘VICE-CHANCELLOR’ so that the vowels are together ?
Answer : 5 × 10 !

Question. How many words ( with or without dictionary meaning) can be made from the letters in the word MONDAY, assuming that no letter is repeated, if
(i) 4 letters are used at a time ?
(ii) all letters are used at a time ?
(iii) all letters are used but the first letter is a vowel ?
Answer : (i) 360 (ii) 720 (iii) 240

Question. In how many ways can 18 white and 19 black balls be arranged in a row so that no two white balls may be together ? It is given that balls of the same colour are identical.
Answer : 190

Question. A customer forgets a four-digit code for an Automatic Teller Machine (ATM) in a bank. However, he remembers that this code consists of digits 3, 5, 6 and 9. Find the largest possible number of trials necessary to obtain the correct code.
Answer : 24

Question. There are 6 items in column A and 6 items in column B. A student is asked to match each item in column A with an item in column B. How many possible answers (correct or incorrect) are there to the question?
Answer : 720

Question. In how many distinct ways can the product xy2z2 be written without using exponents?
Answer : 30

Question. There are two books each of three volumes and two books each of two volumes. In how many ways can the ten books be arranged on a table so that the volumes of the same book are not separated.
Answer : 4 ! 3 ! 2 ! 2!

Question. A library has 5 copies of one book, 4 copies of each of 2 books, 6 copies of each of 3 books and single copies of 8 books. In how many ways can all books be arranged so that copies of the same book are always together?
Answer : 14

Question. There are 3 white, 4 red and 1 blue marbles in a bag. There are drawn one by one and arranged in a row. Assuming that all the 8 marbles are drawn, determine the number of different arrangements if marbles of same colour are indistinguishable.
Answer : 280

Question. Find the number of permutations of the letters of the word PRE-UNIVERSITY.
Answer : 19958392

Question. In how many ways can the letters of the word ‘CIVILIZATION’ be rearranged?
Answer : 13! / 2!2!2!

Question. How many different words can be formed with the letters of the word ‘UNIVERSITY’ so that all the vowels are together.
Answer : 60480

Question. In how many ways can 8 Indians, 4 Americans and 4 Englishmen be seated in a row so that all persons of the same nationality sit together.
Answer : 3 ! 8 ! 4 ! 4!

Question. There are 20 books of which 4 are single volume and the other are books of 8, 5 and 3 volumes respectively. In how many ways can all these books be arranged on a shelf so that volumes of the same book are not separated.
Answer : 7 ! 8 ! 5 ! 3!;

Question. In how many ways can the letters of the word ‘DIRECTOR’ be arranged so that the three vowels are never together?
Answer : 18000

Question. There are 6 candidates contesting for a certain office in a municipal election. In how many ways can their names be listed on a ballot paper ?
Answer : 720

Question. Find the number of rearrangements of the letters of the word ‘BENEVOLENT’. How many of them end in L?
Answer : 302399; 30240

Question. How many different words can be formed with the letters of the worked ‘PENCIL’ when vowels occupy even places.
Answer : 144

Question. How many different words can be formed with five given letters of which three are vowels and two are consonants, no two vowels being together in any word?
Answer : 12

Question. How many three digit numbers with distinct digits are there whose all the digits are odd?
Answer : 60

Question. (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

Question. How many 4-digit numbers greater than 2300 can be formed with the digit 0, 1, 2, 3, 4, 5 and 6, no digit being repeated in any number?
Answer : 560

Question. A number of four different digits without repetition is formed by using the digits 1, 2, 3, 4, 5, 6, 7 Find
(i) How many such numbers can be formed?
(ii) How many of them are greater than 3400 ?
Answer : (i) 840 (ii) 560

Question. Find the number of numbers of 4 digits without repetition formed with the digits 1 2, 3, 4, 5 in which 4 occurs in the thousands place and 5 occurs in the units place.
Answer : 6

Question. (i) How many 2-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, if the digit can be repeated?
(ii) How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?
(iii) How many 5-digit number can be formed using the digits 0, 1, 2, 3, and 4 if the digits can be repeated in a number?
Answer : (i) 10 (ii) 108 (iii) 2500

Question. 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. Ali has 2 school bags, 3 tiffin boxes and 2 water bottles. In how many ways can he carry these items choosing one each.
Answer : 12

Question. Find n, if ^n-1P₃ : ⁿP₄ = 1 : 9 .
Answer : n = 9

Question. How many five digit telephone numbers can be constructed using the digits 0 to 9.
(i) If each number starts with 59, for example 59612 etc., and no digit appears more than once?
(ii) If each number starts with 67 and no digit appears more than once?
Answer : (i) 336 (ii) 336

Question. (i) Given 5 flags of different colours how many different single can be generated if each signal requires the use of 2 flags, one below the other?
(ii) Given 4 flags of different colours, how many different signals can be generated if a signal requires the use of 2 flags one below the other?
(iii) Find the number of different signals that can be generated by arranging at least two flags in order (one below the other) on a vertical staff, if five different flags are available.
Answer : (i) 20 (ii) 12 (iii) 320

Question. (i) How many numbers between 100 and 1000 can be formed with the digits, 1, 2, 3, 4, 5, 6, 7; no digit being repeated?
(ii) How many numbers lying between 100 and 1000 can be formed with the digits 0, 1, 2, 3, 4, 5 if the repetitions of the digits in not allowed?
Answer : 81. (i) 210, (ii) 100

Question. How many numbers each lying between 100 and 1000 can be formed with the digits 2, 3, 4, 0, 8, 9; no digit being repeated?
Answer : 100

Question. Find the total number of ways in which n distinct objects can be put into two different boxes.
Answer : 2ⁿ

Question. A telegraph has 5 arms and each arm can have three distinct positions, including the position of rest. How many signals can be made using the telegraphs?
Answer : 242

Question. 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

Question. How many three digit numbers are there such that at least one of their digit is 7?
Answer : 252

Question. A team consists of 5 boys and 4 girls. It plays single matches against another team consisting of 6 boys and 3 girls. How many matches can be arranged between the two teams if a boys plays against a boy and a girl plays against a girl?
Answer : 42

Question. Rajeev has 3 pants and 2 shirts. How many different pairs of a pant and a shirt, can he dress up with?
Answer : 6

Question. Find r, if 5. ⁴P_r = 6.⁵P_r-1 .
Answer : r = 3

Question. Ten different letters of an alphabet are given, words with five letters are formed from these given letters. Find the number of words which have at least one letter repeated.
Answer : 10⁵ – ¹⁰P₅

Question. In telegraph communication, the Morse code is used in which all the letters of the English alphabet, digit 0 to 9 and even the punctuation marks, all usualy referred as characters, are represented by ‘dots’ and ‘dashes’.
For example, E is represented by a dot (.), T by a dash (–), O by three dashes (– – –), S by three dots (. . .) and so on. Thus, SOS is represented by (. . . – – – . . .) .
(i) How many characters can be transmitted using one symbol (do or dash), two symbols, three symbols, four symbols? Also find the total number of characters which can be transmitted using at most four symbols.
(ii) How many characters can be transmitted by using (a) exactly five symbols? (b) at most five symbols?
Answer : (i) 30 (ii) (a) 32 (b) 62

Question. If ⁿP₅ = 42. ⁿP₃, n > 4, then find the value of n.
Answer : n = 10

Question. If ⁿP₃ = 9240, find n.
Answer : n = 22

Question. Seven songs are to be rendered in a programme. In how many different orders could they be rendered?
Answer : 5040

Question. Ten horses are running a race. In how many ways can these horses come in the first, second and third place, assuming no ties?
Answer : 720

Question. A person is to walk from A to B. However, he is restricted to walk only to the right of A or upwards of A, but not necessarily in the order. One such path is shown in the figure. Determine the total number of paths available to the person from A to B.

Answer : 126

Question. Find the number of words which can be made using all the letters of the word AGAIN. If these words are written as in a dictionary, what will be the fiftieth word?
Answer : NAAIG

Question. Find the number of words that can be made by arranging the letters of the word ‘INTERMEDIATE’ so that
(i) The relative order of vowels and consonants do not change.
(ii) The order of vowels do not change.
Answer : (i) 6! / 2!3! × 6! / 2!3! = 21600
(ii) 12! / 6!2! = 332640

Question. In how many permutations of the letters of the word ‘PARALLEL’ all the L do not come together.
Answer : 3000