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Evaluate )
1)¹⁵C₃
2) ¹²C₉.
3) ⁵⁰C₄₇
4) ⁷¹C₇₁
5) ⁿ⁺¹Cₙ
6) ⁶ᵣ₌₁∑¹⁵Cᵣ
7) ⁸C₄+⁸C₃
8) ¹⁷C₄ +¹²C₃
Simplify
9) If ⁿC₇ =ⁿC₅ find n.
10) If ⁿC₁₄ = ⁿC₂₈ Find ⁿC₂₈
11) If ⁿC₁₆ = ⁿC₁₄ find ⁿC₂₇
12) if ²⁰Cᵣ=²⁰Cᵣ₊₂ find ʳC₅
13) If ¹⁸Cᵣ=¹⁸Cᵣ₊₂ find ʳC₅
14) If ⁿPᵣ=1680 and ⁿCᵣ =70 find n, r
15) If ²ⁿC₃ : ⁿC₃ =11 : 1 find n
16) If ¹⁵Cᵣ : ¹⁵Cᵣ₋₁ = 11: 5 find r.
17) If ⁿCᵣ₋₁ = ⁿC₃ᵣ find r.
18) ⁿ⁺¹Cᵣ₊₁ : ⁿCᵣ = 11 : 6 and
ⁿCᵣ : ⁿ⁻¹Cᵣ₋₁ =6 : 3 find n and r.
19) ⁿCᵣ /⁽ⁿ⁻¹⁾ C₍ᵣ₋₁₎
20) ⁽ⁿ⁻¹⁾C₍ᵣ ₋₁₎ + ⁽ⁿ⁻¹⁾Cᵣ
21) ⁿCᵣ + 2. ⁿCᵣ₋₁ + ⁿ Cᵣ₋₂
22) ¹⁸Cᵣ=¹⁸Cᵣ₊₂ Find r
23) ²ⁿCᵣ = ²ⁿCᵣ₊₂ find r.
24) ⁿC₁₂ = ⁿC₁₀ find ⁿC₂
25) ⁿPᵣ = 336 and ⁿCᵣ=5, find n and r Find also ⁿ⁺² Cᵣ₊₁
26) ²⁰C₃ₙ = ²⁰C₂ₙ₊₅ Find n
27) ⁿC₁₀= ⁿC ₁₅ find ²⁷ Cₙ
28) ²ⁿC₃ : ⁿC₂= 12:1 find n.
29) ¹⁵C₃ᵣ = ¹⁵Cᵣ₊₃ Find the value of r.
30) ²ⁿCᵣ=²ⁿCᵣ₊₂ Find r.
31) ⁿPᵣ=6 . ⁿCᵣ Find n and r when ⁿCᵣ=56.
32) find ⁴⁷C₄+⁵ᵣ₌₁∑⁵²⁻ʳC₃.
33) Evaluate ⁴⁷C₄+²⁰C₅+⁵ᵣ₌₁∑⁵²⁻ʳC₃
34) ⁿC₄ , ⁿC₅ , ⁿC₆ are in A. P then what is the value of n
35) ¹³C₆ + 2. ¹³C₅ + ¹³C₄ =¹⁵Cₓ find x.
36) How many different committee of 5 members may be formed from 8 Indians and 4 foreigners. 792
37)a) In an examination paper 11 questions are set; In how many ways can you choose 6 questions to answer? 462
b) In above question, question number 11 is made compulsory, in how many ways can you select to answer 6 questions in all. 252
38) In how many ways can a committee of 6 men and 2 women be formed out of 10 men and 5 women ? 2100
39) A committee of 10 gentleman and 8 ladies. How many different subcommittee can be formed consisting of 5 gentleman and 3 ladies. 14112
40) In how many ways can a person choose one or more of the four electric appliances? T. V, Refrigerator, Washing machine, Radiogram ? 15
41) A man has 8 friends. In how many ways may be invite one or more of them to a dinner ? 255
42) In how many ways a man can invite 5 friends to a dinner so that two or more of them remain present. 26
43) From 6 boys and 4 girls 5 are to be selected for admission for a particular course, in how many ways can this be done if there must be exactly 2 girls? 120
44) A person has got 12 acquaintances of whom 8 relatives. In how many ways can he invite 7 guests so that 5 of them may be Relatives ? 336
45) At an election there are 5 candidates and 3 members are to be elected and a voter is entitled to vote for any number to be elected. In how many ways may a voter choose a vote? 25
46)a) Find out number of ways in which a cricket team consisting of 11 players can be selected out of 14 players ? 364
b) Find out how many of these will include a particular player.(from above question) 360
** In how many ways can a selection of 6 books be made from 10 books ?
47)a) When one specified book is always included 126
b) When one specified book is always excluded. 84
** In how many ways can a committee of 5 members be formed from 10 candidates so as to
48)a) Include both the youngest and oldest candidates. 56
b) Excluded the youngest if it includes the oldest ?
*** In how many ways can four students be selected out of twelve students if
49)a) 2 Particular students are not included at all. 210
b) 2 particular students are included. 44
50) In an examination paper there are two groups each containing 7 questions. A candidate is required to attempt 9 questions but not more than five questions from any group. In how many ways can 9 questions be selected ? 1470
51) A candidate is required to answer 6 out of 12 questions, which are divided into two groups each containing 6 questions and he is permitted to attempt not more than four from any group. In how many different ways can he make up his choice. 850
52) From 6 gentleman and 4 ladies, a committee of 5 members is to be formed. In how many ways can this be done so as to include atleast one lady ? 246
53) In how many ways can a committee of 5 be formed from 4 professors and 6 students so as to include atleast 2 professor ? 186
54) In an examination paper, there are 14 questions divided into three groups of 5,5,4 respectively. A candidate is required to answer 6 questions taking atleast two questions from each of the first two groups and one question from the third group. In how many ways can he make up his choice ? 1400
55) A committee of 7 members is to be chosen from 6 Chartered Accountants, 4 Economist and 5 Cost Accounts. In how many ways can this be done if in the committee, there must be atleast one member from each group and atleast 3 Chartered Accounts ? 3570
56) There are three sections in a question paper contains 5 questions each. A candidate has to solve any 5 questions choosing atleast one question from each section. In how many ways can be make his choice. 2250
57) In how many ways can a committee of 3 ladies and 4 gentleman be appointed from a meeting consisting of 8 ladies and 7 gentleman? What will be the members of ways If Mrs. X refuses to serve in a committee if Mr. Y is a member ? 1960, 1540
** Out of 4 officers and 10 clerks in an organisation, a committee of five consisting of 2 officers and 3 clerks is to be formed. In how many ways can this be done if
58)a) Any officers and any clerk can be included. 720
b) 1 particular clerk must be on the committee. 216
c) 1 particular officer can't be on the committee. 360
59) The question paper on Mathematics and Statistics contain 10 questions divided into two groups of 5 questions each. In how many ways can an examiner select 6 questions taking atleast two questions from each group? 200
60) Out of six teacher and four boys, a committee of eight is to be formed. In how many ways can this be done when there should be not less than four teachers in the committee. 45
61) An examination paper consists of 12 questions Divided into two parts A and B. Part A contains 7 questions and part B contains 5 questions. A candidate is required to answer 8 questions selecting atleast 3 questions from each part. In how many maximum ways can be selected the question ? 420
62) In a class of 16 students, there are 5 lady students. In how many ways can 10 students be selected from them so as to include atleast 4 lady students. 2772
** In how many ways can 7 men be selected from 16 men so that
63)a) 4 particular men will not be there. 792
b) 4 particular men will always be there. 220
** In a group of 15 boys there are 7 bouy-scouts. In how many ways can 12 boys be selected so as to include
64)a) exactly 6 boys-scouts. 196
b) atleast 6 boys-scouts. 252
65) A cricket team consisting of 11 players is to be selected from 2 groups consisting of 6 and 8 players respectively. In how many ways can the selection be made on the supposition that the group of six shall contribute no fewer than 4 players. 344
64) A cricket eleven is to be chosen from 13 players of whom only 4 can bowl. In how many ways can the team be selected so as to include atleast 2 bowlers ? 78
65) A cricket team of 11 players is to be formed from 20 players including 6 bowlers and 3 wicketkeepers. In how many ways can a team be selected so that the team contains exactly 2 wicketkeeper and atleast 4 bowlers.
27225
66) How many sub-committees of five members can formed from a committee consisting of 8 gentleman and 6 ladies so as to include atleast 2 gentleman and 2 ladies ? 1400
67) A certain council consists of a chairman, two vice-chairman, twelve other members. How many ways can committee of 6 can be formed including always the chairman and only one vice-chairman ? 990
68) A question paper contains 12 questions, of which 7 are in Group A and 5 in Group B. The questions are serially numbered from 1 to 12. If a candidate has to answer the fourth questions and 3 other from Group A and to answer eighth question and 2 other from Group B. In how many ways can the candidate make up his choice. 120
69) Ten electric bulbs, of which 3 are defective are to be tried in three different points in a dark room. In how many ways the total trial the room shall be lighted ? 119
70) There are 10 lamps in a hall. Each of them can be switched on independently. what is the number of ways in which the hall can be illuminated. 1023
71) Five bulbs of which three are defective are to be tried in two light points in a dark room. In how many trials the room shall be lighted. 7
** A supreme court Bench consists of five judges. In how many ways the benches can give
72)a) a majority decision. 16
b) nagation not affecting the majority decision ? 15
73) There are 25 candidates, which include 5 from the scheduled castes for 12 vacancies. If 3 vacancies are reserved for scheduled castes candidates and the remaining vacancies are open to all, find the number of ways in which the selection can be made. ⁵C₃.²²C₉
***There are 16 points on a plane. no three points being on the same line. How many
74)a) Straight line. 120
b) Triangle. 119
c) quadrilateral
d) Pentagon may formed.
75) Find the member of straight lines formed by joining 10 different points on a plane, no three of them being collinear (with the exception of 4 points which are collinear). 40
76) Find the number of triangles formed by joining 10 different points on a plane, no three of them being collinear (with the exception of 4 points which are collinear). 120
** There are 12 points in a plane, of which 5 are collinear. Find
77)a) number of Triangles. 240
b) number of straight lines. 57
78) If 20 straight lines be drawn in a plane, no two of them being parallel and no three of them being concurrent. How many points of intersection will there be ? 190
** Find the number of diagonals in
79)a) Quadrilateral. 2
b) Pentagon. 5
c) Hexagon 9
d) Octagon. 20
e) Decagon 35
80) A polygon has 44 diagonals. Find the number of its sides. 11
81) A man has five friends. In how many ways can he invite one or more of them to a dinner. 31
** A man has 5 German, 4 Spanish and 3 French friends. Find:
82)a) Total ways in which all the friends can be invited. 4096
b) Total ways of invitation so that there is atleast 1 German friend. (2⁵-1)2⁷
c) Atleast 1 German and 1 French friend. (2⁵-1)(2³-1)2⁴
d) Atleast one friend from each country. 3165
83) In an examination paper 10 questions are set. In how many ways can a candidate select one or more questions to answer ? 1023
84) In an examination paper 6 questions are set and every question has an alternative. In how many ways can the questions be answered. 3⁶
85) A league match in football ends in a win,loss,or draw, how many ways forecast can give exactly 18 correct results out of 22 matches played ? ²²C₁₈ .2⁴
86) In an examination a candidate is required to pass four different subjects. In how many ways can he fail ? 15
87) Out of 7 Consonants and 4 vowels, how many words can be formed, each containing 3 consonants and 2 vowels? 25200
88) In how many ways can the letters of the word FORECAST taken 3 at a time and the word MILKY taken 2 at a time be arranged? 67200
** How many different words can be formed taking 4 letters from the following words?
89) a) PROPORTION 758
b) EXAMINATION. 2454
c) EXPRESSION. 2454
d) MATHEMATICS. 2454
e) INEFFECTIVE. 1422
f) MORADABAD. 626
90) In how many ways is it possible to make a selection by taking any number of all 20 fruits, namely 9 mangoes, 7 oranges and 4 apples ? 399
** There are 8 mangoes,4 apples and 5 oranges. If any number of fruits can be selected, find:
91)a) Total Number of selections.
269
b) Total Number of selections so that there is at least 1 mango. 275
c) At least 1 mango and 1 apple. 192
d) At least 1 fruit of each type. 160
*** How many selections can be made by taking any letters from the words.
92) DADDY DID A DEADLY DEED. 1919
93)SHE SELLS SEA SHELLS. 1049
*** There are 3 different green dice, 4 different blue dice and 5 different red dice. If any number of dice can be selected then find.
94)a) total number of selections. 2¹²-1
b) there is atleast 1 green dice.
(2³-1)2⁹
c) there is atleast 1 green 1 blue dice. 105×2⁵
d) there is atleast 1dice of each colour. 3255
95) divide 12 items in two groups so that each containing 8 and 4 items. 12!/(8!4!)
** In how many ways can 12 apples be divided equally among
96)a) 2 boys. 12!/(6!)²
b) 3 boys. 12!/(4!)³
c) 4 boys. 12!/(3!)⁴
d) 6 boys. 12!/(2!)⁶
*** In how many ways can 20 books be equally divided amongst:
97)a) 2 boys. 20!/(10!)²
b) 4 boys. 20!/(5!)⁴
c) 5 boys. 20!/(4!)⁵
d) 10 boys. 20!/(2!)¹⁰
98) The three different sections of a Library need the services of 3,4,5 workers respectively. If 12 workers are available. In how many ways can they be allotted to different sections. 27620
*** In how many ways can 9 students be divided
99)a) equally into 3 groups. 280
b) 3 groups as 2,3,4.
100) A Mathematics paper is divided into three sections. A, B, C consisting of 6,7 and 8 questions respectively. Four questions are to be answered from each section. Find the number of total ways of selecting the questions from the paper by a student. 186
101) there are 5 copies of University physics, 8 copies of University chemistry, 3 different books on law and 2 different books on History in a library. Find the number of ways in which one or more than one book can be selected. 1727
102) There are 5 green dyes, 4 different dyes and 3 different red dyes. How many combination of dyes can be chosen taking atleast one green and one blue dye? 600
103) How many selection can a child make 4 milk chocolates, 5 sugar candies and 6 chewing gums? 209
104) A person has in his bag 14 notes of ₹10 each, 9 notes of ₹5 each, 4 notes of ₹2 each and 7 notes of ₹1 each. In how many different ways can he contribute to a charitable fund ? 5999
105) there are 6 pairs of gloves of different sizes. In how many ways can you choose one for the left hand and one for the right hand such that they are not the same pair? 30
106) in how many ways can you 9 coins of different values into 2 packets ? 512
107) In a test 10 multiple choice questions of one correct answer, each having four alternative answers, then the number of ways to put ticks at random for the answer to all question is. 4¹⁰
108) the number of ways of painting the faces of a cube with six different colour is. 6
109) 8 points are marked on a circle. how many chords can you obtain by joining them in pairs? 28
110) In how many ways can the Supreme Court give a six to three upholding a Lower court? 84
111) taking data from the previous question, in how many ways can it give majority decision reversing the Lower court ? 256
112) There are 10 professor and 20 students out of whom a committee of 2 professors and 3 Students is to be formed. find
a) find the number of ways in which this can be done. 51300
b) find in how many of these committees a particular professor is included. 10260
c) find in how many of these committee a particular student is included. 7695
d) Find in how many of these committees a particular student is excluded. 43605
113) In a football Championship, 153 matches were played. Every team played one match with each other. The number of teams participating in the champions are: 18
114) the number of ways in which a mixed double game can be arranged 9 married couples, if no husband wife play in the same game is . 1512
115) A Captain and vice captain are to be chosen out of a team having 11 players. how many ways are there to achieve this? 110
116) in how many ways can you choose 6 cards out a pack of 52 cards and have all suits present? (13)⁴. ⁴⁸C₃
117) "A number lock" consists of 3 Rings, each marked with 10 different numbers. In how many ways the lock cannot be opened? 999
118) how many different factors can 210 have ? 15
119) how many different factors can 2160 have? 39
120) There is a multiple choice question paper consisting of 6 questions. Each question has four choices. In how many ways can a student attempt one or more question of the question paper ?(the student is a not allowed to mark more than one choice for any question). 5⁶ - 1
121) In how many ways can 4 prizes be given to:
a) 3 boys. 3⁴
b) 4 boys. 4⁴
c) 5 boys. 5⁴
122) In how many ways can 5 prizes be given to:
a) 4 boys. 4⁵
b) 5 boys. 5⁵
c) 6 boys. 6⁵
123) In how many ways can 8 prizes be given to 3 boys? 3⁸
124) In how many ways can 3 prizes be given to 8 boys? 8³
125) In how many ways can 4 letters be put into:
a) 4 boxes? 256
b) 5 boxes? 625
c) 6 boxes? 1296
126) In how many ways can 5 letters be put into:
a) 3 boxes? 243
b) 5 boxes? 2480
c) 6 boxes? 3125
127) In how many ways can 20 different dates can any of the 7 days of the week? 20⁷ and 7²⁰
128) In how many ways the results of:
a) 3 matches be declared when each match ends in a win, loss or a draw ? 3³
b) 10 matches be declared when each match ends in a win, loss or a draw ? 3¹⁰
c) 22 matches be declared when each match ends in a win, loss or a draw ? 3²²
129) In how many ways can 10 balls be distributed into 3 boxes? 3¹⁰
130) In how many ways can'n' biscuits be given to 'N' persons? Nⁿ
131) A man has 5 friends. In how many ways can he invite 1 or more for a dinner ? 31
132) 5 coins are tossed. Find the number of ways where atleast 1 head turns up. 31
133) In an examination a candidate is required to pass in 4 different subjects. In how many ways can he fail ? 15
134) In how many ways can 5 rings be worn in 4 fingers? 1024
135) In how many ways can 3 persons enter into 4 hotels if
a) repetition is allowed. 4³
b) repetition is not allowed. P(4,3)
136) How many 4 digit numbers can be formed with the digits 0, 1, 2, 3,......0(repetition is allowed). 9000
137) In how many ways can 10 balls be put into 5 boxes if repetition is allowed. 5¹⁰
138) In how many ways can 4 children have their birthday in any 1 of the 365 days in a year? 365⁴
139) How many 5 digits numbers can be formed with the digits 0 to 9 so that atleast 1 digits is repeated? 62784
140) In a question paper 10 questions are set. In how many ways one can select 1 or more questions to answer? 1023
141) In the above question find the number of ways of selection if every question has an alternative. 59048
142) Find the ranks of the following words if they are arranged alphabetically in a dictionary:
a) ZENITH. 616
b) LAKE. 21
c) CHALK. 32
d) MOTHER. 309
e) WOMAN. 117
f) AGAIN. 7
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