Question Bank

Abhishek Jain

1. One red flag, three white flags and two blue flags are arranged in a line such that I. no two adjacent flags are of the same colour. II. the flags at the two ends of the line are of difference colours. In how many different ways can the flags be arranged? Ans: 6

2. There are three cities: A, B and C. Each of three cities is connected with the other two cities by at least one direct road. If a traveler wants to go from one city (origin) to another city (destinations), she can do so either by traveling a road connecting the two cities directly, or by travelling two roads, the first connecting the origin to the third city and the second connecting the third city to the destination. In all there are 33 routes from A to B (including those via C). Similarly there are 23 routes from B to C (including via A). How many roads are there from A to C directly? Ans: 6

3. For a scholarship, at the most n candidates out of 2n+1 can be seated. If the number of different ways of selection of at least one candidate is 63, the maximum number of candidates for the scholarship is Ans: 3

4. A box contains 20 tickets of identical appearance, the tickets being numbered 1, 2, 3 ., 20. In how many ways can 3 tickets be chosen such that the numbers on the drawn ticket are in arithmetic progression? Ans: 90

5. 6 students appear in an examination. In how many ways can the result be announced? Ans: 64

6. In how many ways 4 boys and 3 girls can be seated in a row so that boys and girls are alternate? Ans: 144

7. What will be the sum of all the four digit numbers that can be made with the digits 0, 1, 2 and 4? Repetition of digits in a number is not allowed? Ans: 45108

8. In how many different ways 3 different rings can be worn in five fingers of a hand? Ans: 210

9. In how many different ways can you paint all six surfaces of a cube with six different colours? One surface is to be painted

with one colour only? Ans: 30

10. How many four digit numbers can be made with the digits 0, 1, 2 and 7 so that at least one of the digits is repeated in every number? Ans: 174

11. How many selections of 10 balls are possible out of 10 identical red balls, 11 identical green balls and 12 identical yellow balls? Ans: 66

12. In how many ways can you select 2 odd numbers and 2 even numbers out of the first 128 whole numbers? Ans: 64C263C2

13. There are 13 couples, 5 single males and 7 single females in a party. Every male kisses every female once but no one kisses his wife. How many kissings took place in the party? Ans: 347

14. There is a square table and 16 chairs are uniformly put along the sides of it. In how many ways can 16 people sit around it? Ans: 415!

15. A parallelogram is cut by two sets of lines parallel to its sides. Find the number of parallelograms thus formed Ans: (a+2C2)2

16. In how many ways can three integers be selected from the set {1, 2, 3 . 37} such that the sum of the three integers is an odd number? Ans: 3876

17. How many odd integers from 1000 to 8000 (inclusive) have distinct digits? Ans: 1736

18. For some positive integer m, how many integers should be taken from the set of first 2m whole numbers to be sure that at least one of them is add? Ans: m+1

19. A palindrome is a number that reads the same left to right as it does from right to left, such as 131. How many 6-digits palindromes are there which are even? Ans: 400

20. How many 15-letter sets can be made using letters from {A, B, C, D} and requiring that there be at least 2 As, at least 3Bs and at least 2Cs? Ans: 165

21. Five distinct pairs of shoes are displaced. In how many different ways can three shoes be selected containing a matched pair? Ans: 40

22. Ten distinguishable balls are distributed into 4 distinct boxes such that a specified box contains exactly 2 balls. Find the number of such distributions Ans: 4538

23. How many even numbers of four digits can be formed with the digits 1, 2, 3, 4, 5, 6 (repetitions of digits are allowed)? Ans: 648

24. The letters of the word LABOUR are permuted in all possible ways and the words thus formed are arranged as in a dictionary. What is the rank of the words LABOUR? Ans: 242

25. A person wants to select 2 toys for this child. One variety of toys has 9 models and another variety has 6 models. In how many ways can he select the 2 toys one from each of the variety? Ans: 54

26. If (2n+1)P(n-1):(2n-1)Pn = 7:5, find n: Ans: 3

27. If 28C2r:24C2(r-4)=225:11, find r: Ans: 7

28. How many motor vehicle registration numbers of 4 digits can be formed with the digits 0, 1, 2, 3, 4, 5 (no digit being repeated)? Ans: 300

29. In how many ways can the letters of the word POSSESS be arranged so that the four Ss are in alternate positions only? Ans: 6

30. In how many waves can a committee of 3 men and 2 women be formed out of a total of 4 men and 4 women? Ans: 24

31. 7C2 + 5C1 + 8C3 - 7C5 - 5C4 - 8C5 = ? Ans: 0

32. The total number of permutation of the word KOLKATA will be Ans: 1260

33. What is the number of diagonals of a regular polygon with 10 sides? Ans: 35

34. A six faced die, an eight faced die and a ten faced die are thrown together. What is the probable number of outcomes? Ans: 480

35. In an entrance test, a candidate is required to attempt a total of four questions which are to be attempted from two

sections each of which contains 5 questions. The maximum number of questions that he can attempt from any section is 3. In how many ways can he answer in the test? Ans: 200

36. How many triangles can be formed by joining the vertices of a heptagon? Ans: 35

37. In a cultural festival, six programmers are to be staged, three on a day for two days. In how many ways the programmers could be arranged? Ans: 720

38. All the odd numbers from 1 to 9 are written in every possible order. How many numbers can be formed if repetition is nor allowed? Ans: 120

39. How many numbers lying between 3000 and 4000 and made with the digits 3, 4, 5, 6, 7 and 8 are divisible by 5? Repetitions are not allowed? Ans: 12

40. In a meeting between two countries each country has 12 delegates. All the delegates of one country shake hands with all delegates of the other country. Find the number of handshakes

possible? Ans: 144

41. In the previous problem if all the delegates shake hand with each other irrespective of the country they belong to then total number of handshakes is Ans: 276

42. Five persons A, B, C, D, E occupy seats in a row such that A and B sit next to each other. In how many possible ways can these five people sit? Ans: 48

43. A, B, C and D are four towns, any three of which are non-collinear. Then, the number of ways to construct three rods each joining a pair of towns so that the roads do not form a triangle is Ans: 24

44. 20 points, 4 coplanar points are given in space. How many tetrahedrons do they determine? Ans: 4845

45. How many numbers divisible by 2 and lying between 50000 to 70000 can be formed, from the digits 3, 4, 5, 6, 7, 8, 9 no digit being repeated in any number? Ans: 360

46. How many diagonals are there in an n-sided polygon (n 3)? Ans: nC2 - n

47. A class in a school has 40 students. Three students of this class are to be selected as class monitor, games in charge and librarian. In how many ways can they be selected, if a student can hold only one position at a time? Ans: 59280

48. Sherry buys 7 novels from a book fair. Merry buys 8 novels from the fair, none of which is common with those bought by sherry. They decide to exchange their books one for one. In how many ways can they exchange their books for the first time? Ans: 56

49. How many 6-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 and 7 so that the digits should not repeat and the second last digit is even? Ans: 720

50. In how many ways can eight directors, vice-chairman and chairman of a firm be seated around a circular table, if the chairman has to sit between the vice-chairman and a director? Ans: 2 (8)!

51. 8 chairs are numbered from 1 to 8. Two women first choose two chairs from those marked 1 to 4 and 3 men select 3 chairs from the remaining. Find the number of possible arrangements Ans: 1440

52. A polygon has 54 diagonals. Find the number of sides Ans: 12

53. How many numbers can be formed from 1, 2, 3, 4, 5 (without repetition), When the digit at the units place must be greater than that in the tens place? Ans: 60

54. The figure below shows the network connecting cities A, B, C, D, E and F. Then arrows indicate permissible direction of travel. What is the number of distinct paths from A to F?

Ans: 10

55. The product of any r consecutive positive integers must be divisible by Ans: r!

56. In how many ways can 3 children in a family have all different birthdays? Ans: 366365364

57. In how many ways can a leap year have 53 Sundays? Ans: 2

58. A telegraph has 5 arms and each arm is capable of four distinct positions including the position of rest. What is the total number of signals that can be made? Ans: 1023

59. On their 10th wedding anniversary a Bengali couple bought 10 different sweets and then distributed it between two of their family friends such that both of them got five sweets each. Find the number of different ways in which this distribution can be done Ans: 252

60. In the country of Utopia the language contains only 4 alphabets. Find the maximum number of words that can be there in the Utopian dictionary if no alphabet can be repeated in a word Ans: 64

61. A company could advertise about its new product in 4 magazines, 3 newspapers and 2 television channels. But in a later move it decided to give advertisements in only 2 of the magazines, 1 of the newspapers and 1 of the TV channels. In how many ways can they advertise their product? Ans: 36

62. The first 5 odd natural numbers are written in every possible order. How many numbers can be formed if no repetition is allowed and what is their sum? Ans: 5!, 6666600

63. In how many ways one or more than one fruit, can be selected from 6 varieties of fruits, given that there are 5 fruits of each variety? (All the fruits of one variety are identical) Ans: 66-1

64. In a staircase there are 4 steps. A person can jump one step, two steps, three steps or all 4 steps. In how many ways can he reach the top? Ans: 8

65. In a global conference there are 16 delegates who are to be seated along two sides of a long table with 8 chairs on each side. Four delegates being having same views wish to sit on one particular side whereas two delegates having views opposite to them wish to sit on the other side of the table. In how many ways can these 16 delegates be seated? Ans: 8P48P210!

66. After group discussion and interview 6 candidates were selected for admission in a college. But unfortunately the number of seats left is 2. So, it was left with the principal to select 2 candidates out of them. In how many ways can he select 2 candidates? Ans: 15

67. In an examination 10 questions are to be answered choosing atleast 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 10 questions be answered? Ans: 266

68. After a conditioning camp of Indian Cricket team, the final team of 11 players out of a total of 15 players is to be decided such that two players Sourav and Sachin is always chosen. Find the total number of ways the final team can be selected Ans: 715

69. There are two parallel line segments AB and CD in a plane. Line AB contains 12 marked points whereas line CD contains 8 marked points. How many triangles can be formed by using these marked points as vertices? Ans: 864

70. In a box there are 5 distinct white and 6 distinct black balls. A person has to pick up two balls from the box such that there is one each of both the colours. In how many ways he can pick the balls? Ans: 30

71. From a class of 12 students 5 are to be chosen for an excursion. But 3 very close friends decide among themselves that either all three of them will go or none of them will go. In

how many ways can the excursion party be chosen? Ans: 162

72. How many different words can be made from the word EDUCATION so that all the vowels are always together? (Do not bother about any meaningless words) Ans: 14400

73. How many numbers are there between 100 and 1000 such that every digit is either 4 or 5? Ans: 8

74. If parallel lines in a intersected by a family of b parallel lines, then find the number of parallelograms formed in this process Ans: aC2 bC2

75. Find the total number of words that can be made by using all the letters from the word MACHINE using only once Ans: 5040

76. A total of 66 games were played in a tournament where each player played one against the rest. The numbers of players are Ans: 12

77. The number of 2 digit even numbers formed from the digits 1, 2, 3, 4, 5 and 6 if repetition of digits is not allowed Ans: 15

78. How many different words can be formed by taking any three letters form the word CHEMISTRY ? Ans: 504

79. How many six lettered words starting with the letter T can be made from all the letters of the word TRAVEL? Ans: 120

80. What is the value of 18C16? Ans: 153