1. In a school 20 students gave gift to each other, then find the total number of gifts.
2. In how many ways at least 6 players can be selected from a group of 8 players.
3. In how many ways Ramesh can invite 6 friends out of his 15 friends for a dinner.
4. In how many ways at least 6 student be selected out of 12 students?
5. In how many ways 10 friends can be invited out of 20 friends for dinner if Ramesh & rahul are always invited.
6. A work can be compelted by 5 men or 7 women in 4 days. In how many ways labour force can be selected if work has to be finished in 4 days & available labour force is 12 men & 10 women? If labour force has only men or only women.A work can be compelted by 5 men or 7 women in 4 days. In how many ways labour force can be selected if work has to be finished in 4 days & available labour force is 12 men & 10 women? If labour force has only men or only women.
7. Out of 5 men & 10 women a team of 6 is to be formed such that at least one men is included in the team. In how many ways such a team can be formed?
8. Out of 4 boys & 6 girls a team of 5 person is to be formed such that the team should have at least 1 boy and two boys named Rahul & Ankit refuse to work together. In how many ways team can be formed?
9. A fruit basket has n identical mangoes, n identical oranges & n distinct fruit. what is the number of ways in which n fruits can be selected from the fruit basket?
10. Find the number of ways in which three numbers in A.P. can be selected from a set of n integers {1, 2, 3, ………n}.
11. What is the number of ways in which three numbers in A.P. can be selected from the given set {1, 2, 3, 4, 5,…………120}?
12. What is the number of ways in which three numbers in A.P. can be selected from the given set {1, 2, 3, 4, 5, ……….. , 121}?
13. Let there are 8 oranges, 6 mangoes & 3 apples
(i) How many selection of fruits can be made?
(ii) How many selection of fruits can be made such that at least 1 apple is always included?
(iii) How many selection of fruits can be made such that at least one fruit of each type is always included?
14. In how many ways 7 players can be selected out of 15 players if Virat is always included?
15. In how many ways at least 5 players can be selected from a group of 7 players?
16. In how many ways a team of p students can be selected from q students such that r student are always included & s students are always included?
17. If a bus conductor has 435 different tickets then findd the number of stoppage that the bus has, consider only one way journey.
18. A student is allowed to select atmost n books from a collection of (2n + 1) books. what is the number of ways of doing this?
19. How many factors of N = 2⁵×3⁵×5⁵×7⁵×11⁵ ends with 3 zeros?
20. In how many ways a team of 11 players can be selected from 18 players if out of 18 players, 2 player can play only as wicket keeper.
21. What is the total number of ways of selecting fruits with at least 1 mango from 5 distinct mangoes & 4 distinct oranges
22. 15 singers were selected to perform in a party, but at a time only 5 singers can perform, then total how many times performance was done if no same group of 5 singers performed twice?
23. 15 singers were selected to perform in a party, but at a time only 5 singers can perform, then how many times a particular singer performed if no same group of 5 singers performed twice?
24. In how many ways 5 number can bee selected from {1, 2, 3, 4, 5, 6, ……….n} such that all the five numbers are cosecutive?
25. In how many ways 5 numbers can be selected from {1, 2, 3, 4, 5, 6, ………. n} such that not all the five numbers are consecutive?
26. If r, s, t are prime numbers & p, q are positive integers such that the LCM of p, q is r²t²s⁴, then the number of ordered pair (p,q) is?
27. In a certain test, aᵢ students gave wrong answer to at least i question where i = 1, 2, 3, ………, k & no student gave more than k wrong answers. What is the total number of wrong answers?
28. In a certain test, 2⁽ᵏ⁻ᶦ⁾ students gave wrong answers to at least i questions, where i = 1, 2, 3, ………., k & no student gave more than k wrong answers. what is the total number of wrong answers given?
29. 30 players from India, Canada & Russia participatedd in world chess tournament. Number of players from india is 3 more than the number of players from canada. In tournament players from same country are not allowed to play against each other & no two player plays against each other more than once. Then find
(i). The maximum number of matches that was played in the tournament
(ii). If number of matches is maximum then what is the ratio of number of players from India & Russia ?
30. A committee of 5 men & 5 women is to be selected from 9 men & 10 women.
(i). In how many ways this can be done
(ii). In how many ways this can be done if a particular woman is always selected
(iii). In how many ways this can be done if a particular woman is always selected and a man is always rejected?
31. In the annual function of a school every student gave a gift to every other student. If total number of gift distributed is 380 then
(i). How many students were present in the function?
(ii). How many gifts received by a particular student Amit ?
32. How many factors of N = 2⁷ × 3⁷ × 5⁷ × 7⁷ ends with two zeros?
33. (i). Find the total number of factors of 54000
(ii). Sum of all factors of 54000
(iii). Sum of all factors of 54000 that ends with 5.
34. There are 10 singers who is ready to perform in a show that is scheduled for two timings, before lunch & after lunch. If before lunch maximum 6 singers can perform while after lunch maximum 5 singers can perform then in how many different way program can be scheduled?
35. In how many ways 5 students from a group of 10 students be selected if Ankit & Rahul are not selected together?
36. In how many ways a team of 4 boys & 5 girls out of total 8 boys & 10 girls can be formed?
37. In how many ways 5 players can be selected out of 10 players
(i). If Ankit & Rahul are always selected
(ii). If Ankit & Rahul are always rejected.
38. Find the number of ways of selecting 11 players from a group of 35 iplayers such that the oldest & the youngest players are always included in the group.
39. In how many ways 5 vowels & 6 consonants can be selected from 26 letters of alphabet?
40. At an election three wards of a town are canvassed by 6, 7 & 10 men repectively. If there are 30 volunteers, in how many ways can they be allotted to different wards?
41. How many different sums can be formed with the following coins:⟶ one 1 ruupee, one 50 paise, one 25 paise, one 10 paise & one 5 paise coins?
42. Given 6 different green dyes, 5 different blue dyes & 4 different red dyes, the number of combination of dyes which can be chosen taking at least one green & one blue dye is?
43. In how many ways 11 players can be selected out of 25 players if Ram & Shyam are always selected and Mohit & Ankit are always rejected ?
44. In how many ways 8 distinct alphabets can be selected so that no vowel is selected?
45. In class (V) there are three sections A, B & C with number of students 50, 70 and 90 respectively while in class (vi) there are three sections A, B & C with number of students 70, 90 & 50 respectively. In how many ways 15 students can be selected either from class V or class VI equal from section A, B & C.
46. In an examination a student has to write 8 papers & has to pass in each paper to pass the exam. In how many ways a student can fail in the exam?
47. In the annual function of a school 10 boys & 20 girls danced such that in a particular dance only 1 boy & 1 girl participated. Total how many dance was performed?
48. In a dance competition, there are 10 participants. In each performance exactly two dancers will perform. Mr. Abhishek, a fan of this show has decided to see exactly 10 dance iperformance. In how many ways he can select the performance?
49. Consider a set x = {1, 2, 3, …….., 9, 10}. What is the number of pair {A, B} such that A ⊆ x and B ⊆ x also A ≠ B and A ∩ B = {2, 3, 5, 7}.
50. A committee of 12 students has to be selected from 18 students 6 from each management, sales & finance such that at least 2 student must be selected from each management, sales & finance and number of students from management is more than the number of students from sales which in turn is more than the number of students from finance. In how many ways this selection can be achieved?
51. In an examination the question paper has three sections namely physics, chemistry & Biology containing 5, 6 & 7 questions respectively. As per the rule a student has to answer minimum 3 questions from each section. In how many ways a student can attempt 10 questions?
52. In an exam a question paper has 10 questions. What is the total number of ways that a student can answer this paper?
53. Find the number of factors of 1800.
54. How many factors of 1800 other than 1800 are 4 digit number?
55. A class has 10 students 7 are boys and 3 are girls. In how many ways students can be selected for a project.
56. In a class of 12 students 7 are boy & rest are girls. In how many ways students can be selected for the project such that at least 1 boy & 1 girl is selected.
57. In a class of 12 students 7 are boys and rest are girls. In how many ways students can be selected for a project such that at least 2 boys & 2 girls are selected?
58. There are 8 questions in an examination. A student has to answer at least four question to pass the exam, in how many ways he can fail in the exam?
59. A fruit basket has 5 bananas, 6 Lemon & 5 different type of fruits. In how many ways one can select fruits from this fruit basket?
60. In a fruit basket 5 mangoes, 6 banana & 4 Lemons are kept. In how many ways one can select fruits from this fruit basket such that at least one fruit of each type is always selected?
61. Let p be the number of factors of 3000 & q be the number of factors of 4000 & r be the number of common factors of 3000 and 4000. Then find the ratio of (p-r):(q-r)
62. If ‘p’ is the number of ways in which three numbers in A.P. can be selected from 1, 2, 3, 4, …….., n. Then find the value of p if
(i). n is odd
(ii). n is even
63. There are infinite number of letters F, L and Y. In how many ways we can choose ‘n’ letters such that the word ‘FLY’ can not be formed?
64. Let E = {1, 2, 3, 4} and F = {1, 2} then the number of onto functions from E to F will be?
65. A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw.
66. A question paper is divided into two parts A & B and each part contains 5 questions. The number of ways in which a candidate can answer 6 questions selecting at least two questions from each part is:
67. Out of 8 men and 4 women a committee of 5 is to be formed. In how many ways this can be done:
(i). So as to include at least 1 woman in the committee
(ii). So as to include at least 1 woman in the committee & exclude at least 1 man from the committee.
68. In how many ways two numbers can be selected from 100, 101, 102, ……….199 such that when these selected numbers are multiplied then the product is divisible by 3?
69. In how many ways can a mixed double game in Tennis be arranged from 20 married couple, if no husband and wife play in the same game.
71. How many 2 digit numbers exist whose number of factors is 8?
72. How many natural numbers less than 1000 will have exactly 3 factors?
73. A fruit basket has ‘k’ different types of fruit and each type has ‘n’ number of fruits. In how many ways at least 1 fruit can be selected from the fruit basket?
74. A team of 11 players has to be selected from 20 players which has 5 bowlers. IN how many ways this can be done if at least 4 bowlers are always selected?
75. A class has total 10 students: 6 boys and 4 girls, in how many ways we can select a team of 5 so that number of boy is always more than the number of girls?
76. How many teams of 9 players an be made from 30 players if Ankit, Mohit and Neeraj can not be selected together?
77. 8 students has to be selected such that 4 students from science group & 4 students from Arts group. Number of participants in different group is as follows:-
Science group:- Total 9 students – 5boys and 4 girls
Art group:- Total 9 students – 4 boys and 5 girls
How many ways group of 8 students can be selected if it has 4 boys and 4 girls?
78. A class has 10 students. In how many ways 4 students can be selected if a particular student amit is selected ‘x’ times and out of those ‘x’ times another student Rohit came ‘y’ times with Amit. By what percentage ‘x’ is more than ‘y’?
79. In the previous question if a prticular student Amit came with Rohit but not with shyamu then find the number of ways to select 4 students out of 10 students.
80. A class has 30 students out of which 23 are boys & 7 are girls. In how many ways 15 students can be selected if 5 seats are reserved for girls student?
81. In an election number of contestants is one more than the number of seats. If a voter can give minimum 1 and maximum equal to the number of seats vote then in how many different ways a particular voter can vote?
(i) 2044
(ii)2045
(iii) 2046
(iv) none of these
82. In the previous question to vote for 1 candidate a voter needs 1 ballot paper then which one of the following could be the total number of ballot paper:
(i). 4401
(ii). 4402
(iii). 4403
(iv). 4405
83. Consider a set s = {1, 2, 3, ………., 200}, two elements p and q are selected from this set s such that 7ᵖ + 7ᑫ is divisible by 5. In how many ways this selection can be done?
84. In an election the number of contestant is one more than the number of seats. A voter can give minimum 1 and maximum equal to the number of seats. Then find the number of contestant if a particular voter can vote in 510 different ways.
85. A committee of 12 is to be formed from 9 women and 8 men.
(i). In how many ways this can be done if at least 5 women have to be included in a committee?
(ii). In how many of these committees women are in majority?
(iii). In how many of these committees the men are in majority?
86. Let A and B be two sets containing 2 & 4 elements respectively. The number of subsets of A×B having 3 or more elements is?
87. In a certain test there are n questions. In this test 2ᵏ students gave wrong answers to at least n-k questions. k = 0, 1, 2, ……n. If the number of wrong answers is 8191 then what is the value of n?
88. A class contains 4 boys and x girls. In a test only 5 students including at least 3 boys can appear. If different group of students appear for the exam every time, if the number of times test conducted is 435 then find the total number of students in the class.
89. In a school, roll no. of k students are given from 1 to k. Three students are selected from these k students such that their roll numbers are not consecutive. In how many ways selection can be made?
90. A student is allowed to select at most n books from a collection of (2n+1) books. If the total number of ways in which he can select at least one book is 63, then what is the value of n?
91. 7 relative of a man comprises 4 ladies and 3 gentlmen; his wife has also 7 relatives; 3 of them are ladies & 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of man’s relative and 3 of the wife’s relative?
92. A set S has n elements. what is the number of subsets of this set which contains odd number of elements?
93. 8 people are to be selected for a work from a group of 20 people such that exactly one from Mr. & Mrs. A is selected, similarly one from Mr. & Mrs. B and Mr. & Mrs. C and Mr. & Mrs. D is selected for the work. How many ways this can be done?
94. In an election 3 person are to be selected from 6 candidates. A voter can cast any number of votes but not more than the candidates to be elected. In how many ways can he cast his vote?
