PERMUTATIONS AND COMBINATIONS
KEY POINTS * When a job is performed in different ways then each way is called the permutation. * Fundamental principle of counting: If a job can be performed in m different ways and for each such way, second job can be done in different ways, then the two jobs can be completed in m x n ways. * Fundamental principle of Addition: If there are two events such that they can be performed independently in m and n ways respectively, then either of two events can be performed in (m +n) ways. * The number of arrangements ( permutations ) of n different things taken r at times is * The number of selections ( Combinations ) of n different things taken r at time is * , * No. of permutations of n things, taken all the time, of which p are alike of one kind, q are alike of 2nd kind such that p + q=n, is * Total no. of ways in which one or more things out of n different things can be selected in ways. * * * -------------------------------------------------

Practice questions 1) a) How many numbers are there between 100 and 1000 such that 7 is in the unit’s place? Repetitions of the digits is allowed)
b) How many numbers are their between 100 and 1000 such that at least one of their digit is 7?
c) How many of them have exactly one of their digits as 7? 2) If there are 20 steamers playing between places A and B, I how many ways could the round trip from B to A, be made if the return was made on i) same steamer; ii) Different steamer. 3) In how many ways ca 5 persons sit in the car, 2 including the driver in the front seat and 3 in the back seat, if 2 particular person out of the 5 are to avoid the driver’s seat? 4) A class consists of 40 girls and 60 boys. I how many ways can a president, vice-president, treasurer and secretary be chosen if the treasurer must be a girl, the secretary must be a boy and a student may not hold more than one office? 5) 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. 6) There are 6 multiple choice questions in an examination. How many sequences of answer are possible if the first three questions have 4 choices and the next three have 5 each? 7) a) Six pictures are to be arranged from left to right on a wall of an art gallery for display. How many arrangements are possible?
b) Find the number of 4-letter words, with or without meaning, which can be formed out the letters of the word ROSE, where the repetition of the letter is not allowed. 8) For a group paragraph, 3 boys and 2 girls stand in a line in all possible ways. How many photos could be taken in each photo corresponds to each such arrangements? 9) Twelve students compete in a race. In how many ways can be first three places be taken? 10) How many integers between 1,000 and 10,000 have no digits other than 4, 5 or 6? 11) Four tourists have a choice of 5 hotels in a city. Each wants to stay in a different hotel. Find the number of ways in which they can make their choice? 12) Eight children are to be seated on a bench:
a) In how many ways can the children be seated?
b) How many arrangements are possible if the youngest child sits at the left hand end of the bench? 13) A sample of 5 bulbs is tested. A bulb is labeled as ‘G’ if it is good and ‘D’ if it is defective. Find the number of all possible outcomes. 14) In Delhi, telephone numbers consists of 6 digits and none of them begins with 0. How many could be possible in Delhi? 15) In a certain city, all telephone numbers have six digits, the first two digits always being 4146 or 62 or 64. How many telephone numbers have all six digits distinct? 16) How many numbers divisible by 5 and lying between 4000 and 5000 can be formed from the digit 4, 5, 6, 7 and 8? 17) A customer forgets a four digit code for an Automatic Teller Machine in a bank. However, 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. 18) Three diced are rolled. Find the number of possible outcomes in which at least one die shows 5. 19) If are the ratio in 2:1, find the value of n. 20) Find the value of n if: 21) Prove that ….. 22) Prove that. 23) Find the value of n such that:
a)
b) 24) How many different numbers can be formed from the digits 1,2,3,4 and 5 assuming that the digits cannot be repeated? How many of these will be even? 25) In how many ways 5 boys and 3 girls be seated in a row so that no two girls are together. 26) Find the number of permutations of the letters of the word KAPIL.
Beginning with K and ending with L.
P being always in the middle.
Vowels being always together.
Vowels being never together.
Vowels occupying odd places. 27) Three married couples are to be seated in a row having six seats in a cinema hall. If spouse are to be seated next 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. 28) The principle wants to arrange 5 students on the platform such that the boy SALIM occupies second positions and such that the girl SITA is always adjacent to the girl RITA. How many such arrangements are possible? 29) The code word is to consist of two distinct English alphabets followed by two distinct numbers from 1-9. For example AK47 is a code word. How many such code words are there. How many of them end with an even integer? 30) Find the number of 4 digit numbers that can be formed using the digits 1, 2, 3, 4 and 5 if no digit is repeated. How many of these will be even? 31) How many different numbers can be formed from the digits 2,3,5,7 and 9? How many of them are odd? (Assuming no digit repeat). 32) 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:
End in a 4?
End in a 3?
End in a 3 or 6? 33) The letters of the word TUESDAY are arranged in a line, each arrangement ending with the letter S. How many different arrangements are possible? How many of them start with the letter D? 34) Find the number of words that can be formed with the letters of the word DELHI, beginning with D, L being always in the middle. vowels being always together, vowels being never together. 35) How many words can be formed out the letters of the word ARCTICLE so that the vowels may occupy even places. 36) How many words can be formed out the letters of the word ORIENTAL so that A and E always occupy odd places? 37) Find the number of arrangements of letters of the word SUBJECTION when the relative position of vowels and consonants are not changed? 38) Ten horses are running a race. In how many ways can these horses come in the first, second and third place, assuming no ties? 39) In how many ways can 5 girls and 3 boys be seated in a row so that no two boys are together? 40) In how many ways 4 boys and 3 girls be seated in a row on 7 chairs if boys and girls alternate? 41) The of a collage consists of one representative from each of six classes, first year to sixth year. In how many ways can they be seated in a row so that the first year and sixth year representatives may be:
Always together? Never together? 42) In how many ways can 9 examination papers be arranged so that the best and the worst papers never come together? 43) 7 candidates are to take examination, 2 in Mathematics and the remaining in different subjects. In how many ways can they be seated in a row so that the two examinees in mathematics may not sit together? 44) In how many ways can 8 sweets of different sizes be disturbed among 8 boys so that the largest sweet goes to the youngest boy? 45) Using the digits 1,2,3,4,5 how many 3 digit numbers ( without re[eating the digits) can be made? 46) In how many ways 7 pictures can be hanged on 9 pegs? 47) Ten buses are plying between the two places A and B, In how many ways a person can travel from A to B and come back? 48) There are 10 non collinear points in a plane. By joining them how many chords can be drawn? 49) There are 10 points on a circle. By joining them how many triangles can be made? 50) If, find x. 51) If, find x. 52) How many different words (with or without meaning) can be made using all the vowels at the time? 53) Using 1,2,3,4,5 how many numbers greater than 10000 can be made? (Repetition not allowed) 54) If. 55) In how many ways 4 boys can be chosen from 7 boys to make a committee? 56) How many different words can be formed by using all the letters of the word SCHOOL? 57) In how many ways can the letters of the word PENCIL be arranged so that I is always next to L? 58) In how many ways 12 boys can be seated on 10 chairs in a row so that two particular boys always take seat? 59) In how many ways 7 positive and 5 negative signs can be arranged in a row so that no negative sign occur together? 60) From the group of 7 boys and 5 girls, a team consisting of 4 boys and 2 girls is to be made. In how many ways it can be done? 61) In how many ways can one select a cricket team of eleven players from 17 players in which only 6 players can bowl and exactly 5 bowlers are to be included in the team? 62) In how many ways 11 players can be chosen from 16 players so that 2 particular players are always excluded 63) Using the digits 0, 1, 2,2,3 how many numbers greater than 20000 can be made? 64) If the letters of the word PRANAV are arranged as in dictionary in all possible ways, then what will be 182nd word? 65) From a class of 15 students, 10 are to chosen for a picnic. There are two students who decide that either both will join or none of them will join. In how many ways can the picnic be organized? 66) Using the letters of the word ARRANGEMENT how many different words can be made such that both A and E both R and both N occur together? 67) A polygon has 35 diagonals. Find the number of its side? 68) How many products can be obtained by multiplying two or more of the numbers 2,3,6,7,9? 69) Determine the number of 5 cards combinations out of pack of 52 cards if at least 3 out of 5 cards ace cards? 70) How many words can be formed from the letters the word ORDINATE so that the vowels occupy odd places? 71) Using the digits 0,1,2,3,4,5,6 how may 4 digit even number can be made, no digit being repeated? 72) There are 15 points in plane out of which only 6 are in a straight line, then 73) a) How many different straight lines can be made?
b) How many triangles can be made? 74) If there are 7 boys and 5 girls in a class, then in how many ways they can be seated in a row such that a) No two girls sit together? b) All the girls never sit together? 75) Using the word EDUCATION how many words using 6 letters can be made so that every word contains at least 4 vowels? 76) What is the number of ways choosing 4 cards from a deck of 52 cards? In how many of these,
3 red and 1 black.
All 4 cards are from different suits.
At least 3 are face cards?
All 4 cards are of the same color? 77) How many 3 letter words can be formed using the letters of the word INEFFECTIVE? 78) How many 5 letter words containing 3 vowels and 2 constants can be formed using the letters of the word EDUCATION so that 3 vowels always occur together? 79) The students in the class are seated according to their marks in previous examination. Once it is so happen that four of these students got equal marks and therefore, they are. To decide their seating arrangements , the teacher wants to wants to write down all possible arrangements, one in each of separate bits of paper in order to chose one of these lots. How many bits of paper are required? 80) In a class, there are 30 boys and 18 girls. The teacher wants to select one boy and one girl to represent the class for a quiz competition. In how many ways can the teacher make this selection? 81) The digits from 0-9 are written on slip of paper and placed in a box. Three of the slips of paper are drawn and placed in order. How many different outcomes are possible? 82) How many numbers can be formed from the digits 2,5,7 and 9 if repetition of the digits is not allowed? 83) How many numbers each less than 1,000 can be formed from the digits 2,5,6,8 and 9 in it. c) No digit is repeated? d) Repetition of digit is allowed? 84) How many 5 digit number each greater than 23,000 can be formed using the digits 1.2.3.6.8? Repetition of digits are not allowed) 85) A die is tossed once and the outcome is recorded. How many possible outcomes are there? What if the die is tossed two times? Three times? n times? 86) If , find n. 87) How many 4 digit numbers can be formed with the digits 1,2,3,4 which are i. odd; ii. Even ; iii greater than 2000; iv. lying between 1000 and 3000; v. divisible by 4? 88) How many of the NATURAL number from 1-1000 have none of their digits repeated? 89) When a group photograph is taken, all the seven teachers should be in the first row and all the twenty students should be in second row. If the two corners of the second row are reserved for the two tallest students, interchangeable only between them, and if the middle seat of the front row is reserved or the principle, how many such arrangements are possible? 90) How many 3 letter words can be using a letter of the word ORIENTAL? 91) How many words (with or without dictionary) can be made from the letters in the word MONDAY, assuming that no letter exactly once. 92) How many odd number greater than 80000 can be formed using digits 2,3,4,5 and 8 if the digit is u used only once in a number? 93) There are three blue balls, four red balls and five green balls. In how many ways they can be arranged in a row? 94) If , determine the value of n? 95) Evaluate

ANSWERS 1) 72 Members; 72 Members; 81 Members 2) A = 20 x 19 = 380 3) 3 x 4 x 2 x 1 = 72 4) NA 5) 320 6) 8000 7) i.720; ii. 24 8) 120 9) 1320 10) 81 11) 120 12) i. 40320; ii.5040 13) 14) 900000 15) 8400 16) 25 17) 24 18) 19) n=0 or 5 20) n=4 or -1/3 21) n=6 22) 23) Ed 24) i. n=10,3 ii. n=10
1 dig=8 2 dig=8 3 dig=24 4 dig=48 25) 14400 26) NA 27) 4x6x6=144 28) 4x2=8 29) i. 1=660 way ii. 20800 30) 120,48 31) 325,260 32) 24 33) 120,48 34) i. 24 ii. 24 iii.48 35) 144 36) 8640 37) 17280 38) 720 39) 14400 40) 144 41) i. 240 ii. 480 42) 282240 43) 3600 44) 5040 45) 60 46) 47) 100 48) 45 49) 120 50) 513 51) N=6 52) 120 53) 120 54) 1 55) 35 56) 360 57) 120 58) 59) 56 60) 350 61) 2772 62) 364 63) 36 64) PANVAR 65) 66) 5040 67) 10 68) 26 69) 4560 70) 576 71) 420 72) i. 93 ii. 435 73) i. ii. 74) 24480 75) G 76) 265 77) All 3 letters are different
2 are identical 1 is different 78) 1080 79) 4x3x2=24 80) 540 81) 720 82) 64 83) 85, 155 84) 90 85) N=3 86) i. 1,3 ii.2,4 iii.2,3,8 iv.2x6=12 v.2 ways 87) 2134 88) 140x18! 89) 336 90) 360, 720, 240 91) 12,100 92) 27720 93) N=14, R=4 94) 142506 95) 190