JEE Mathematics Permutation and Combination MCQs Set 05

Choose the most appropriate option (a, b, c or d).

Question. There are 10 points in a plane of which no three points are collinear and 4 points are concyclic. The number of different circles that can be drawn through at least 3 points of these points is
(a) 116
(b) 120
(c) 117
(d) None of the options
Answer: (c) 117

Question. In a polygon the number of diagonals is 54. The number of sides of the polygon is
(a) 10
(b) 12
(c) 9
(d) None of the options
Answer: (b) 12

Question. In a polygon no three diagonals are concurrent. If the total number of points of intersection of diagonals interior to the polygon be 70 then the number of diagonals of polygon is
(a) 20
(b) 28
(c) 8
(d) None of the options
Answer: (a) 20

Question. n lines are drawn in a plane such that no two of them are parallel and no three of them are concurrent. The number of different points at which these lines will cut is
(a) \( \sum_{k=1}^{n-1} k \)
(b) \( n(n - 1) \)
(c) \( n^2 \)
(d) None of the options
Answer: (a) \( \sum_{k=1}^{n-1} k \)

Question. The number of triangles that can be formed with 10 points as vertices, n of them being collinear, is 110. Then n is
(a) 3
(b) 4
(c) 5
(d) 6
Answer: (c) 5

Question. There are three coplanar parallel lines. If any p points are taken on each of the lines, the maximum number of triangles with vertices at these points is
(a) \( 3p^2(p - 1) + 1 \)
(b) \( 3p^2(p - 1) \)
(c) \( p^2(4p - 3) \)
(d) None of the options
Answer: (c) \( p^2(4p - 3) \)

Question. Two teams are to play a series of 5 matches between them. A match ends in a win or loss or draw for a team. A number of people forecast the result of each match and no two people make the same forecast for the series of matches. The smallest group of people in which one person forecasts correctly for all the matches will contain n people, where n is
(a) 81
(b) 243
(c) 486
(d) None of the options
Answer: (b) 243

Question. A bag contains 3 black, 4 white and 2 red balls, all the balls being different. The number of selections of at most 6 balls containing balls of all the colours is
(a) \( 42(4!) \)
(b) \( 2^6 \times 4! \)
(c) \( (2^6 - 1)(4!) \)
(d) None of the options
Answer: (a) \( 42(4!) \)

Question. In a room there are 12 bulbs of the same wattage, each having a separate switch. The number of ways to light the room with different amounts of illumination is
(a) \( 12^2 - 1 \)
(b) \( 2^{12} \)
(c) \( 2^{12} - 1 \)
(d) None of the options
Answer: (c) \( 2^{12} - 1 \)

Question. In an examination of 9 papers a candidate has to pass in more papers than the number of papers in which he fails in order to be successful. The number of ways in which he can be unsuccessful is
(a) 255
(b) 256
(c) 193
(d) 319
Answer: (b) 256

Question. The number of 5-digit numbers that can be made using the digits 1 and 2 and in which at least one digit is different, is
(a) 30
(b) 31
(c) 32
(d) None of the options
Answer: (a) 30

Question. In a club electron the number contestants is one more than the number of maximum candidates for which a voter can vote. If the total number of ways in which a voter can be 62 then the number of candidates is
(a) 7
(b) 5
(c) 6
(d) None of the options
Answer: (c) 6

Question. The total number of selections of at most n things from (2n + 1) different things is 63. Then the value of n is
(a) 3
(b) 2
(c) 4
(d) None of the options
Answer: (a) 3

Question. Let \( 1 \leq m < n \leq p \). The number of subsets of the set \( A = \{1, 2, 3, \dots, p\} \) having \( m, n \) as the least and the greatest elements respectively, is
(a) \( 2^{n-m-1} - 1 \)
(b) \( 2^{n-m-1} \)
(c) \( 2^{n-m} \)
(d) None of the options
Answer: (b) \( 2^{n-m-1} \)

Question. The number of ways in which n different prizes can be distributed among \( m (< n) \) persons if each is entitled to receive at most \( n - 1 \) prizes, is
(a) \( n^m - n \)
(b) \( m^n \)
(c) \( mn \)
(d) None of the options
Answer: (d) None of the options

Question. The number of possible outcomes in a throw of n ordinary dice in which at least one of the dice shows an odd number is
(a) \( 6^n - 1 \)
(b) \( 3^n - 1 \)
(c) \( 6^n - 3^n \)
(d) None of the options
Answer: (c) \( 6^n - 3^n \)

Question. The number of different 6-digit numbers that can be formed using the three digits 0, 1 and 2 is
(a) \( 3^6 \)
(b) \( 2 \times 3^5 \)
(c) \( 3^5 \)
(d) None of the options
Answer: (b) \( 2 \times 3^5 \)

Question. The number of different matrices that can be formed with elements 0, 1, 2 or 3 each matrix having 4 elements, is
(a) \( 3 \times 2^4 \)
(b) \( 2 \times 4^4 \)
(c) \( 3 \times 4^4 \)
(d) None of the options
Answer: (c) \( 3 \times 4^4 \)

Question. Let A be a set of \( n (\geq 3) \) distinct elements. The number of triplets \( (x, y, z) \) of the elements of A in which at least two coordinates are equal is
(a) \( {}^nP_3 \)
(b) \( n^3 - {}^nP_3 \)
(c) \( 3n^2 - 2n \)
(d) \( 3n^2(n - 1) \)
Answer: (c) \( 3n^2 - 2n \)

Question. The number of different pairs of word ( [Grid] ) that can be made with the letters of the word STATICS is
(a) 828
(b) 1260
(c) 396
(d) None of the options
Answer: (b) 1260

Question. Total number of 6-digit numbers in which all the odd digits and only odd digits appear, is
(a) \( \frac{5}{6}(6!) \)
(b) \( 6! \)
(c) \( \frac{1}{2}(6!) \)
(d) None of the options
Answer: (a) \( \frac{5}{6}(6!) \)

Question. The number of divisors of the form \( 4n + 2 (n \geq 0) \) of the integer 240 is
(a) 4
(b) 8
(c) 10
(d) 3
Answer: (a) 4

Question. In the next World Cup of cricket there will be 12 teams, divided equally in two groups. Teams of each group will play a match against each other. From each group 3 top teams will quality for the next round. In this round each team will play against others once. Four top teams of this round will qualify for the semifinal round, where each team will play against the others once. Two top teams of this round will go to the final round, where they will play the best of three matches. The minimum number of matches in the next World Cup will be
(a) 54
(b) 53
(c) 38
(d) None of the options
Answer: (b) 53

Question. The number of different ways in which 8 persons can stand in a row so that between two particular person A and B there are always two persons, is
(a) \( 60(5!) \)
(b) \( 15(4!) \times (5!) \)
(c) \( 4! \times 5! \)
(d) None of the options
Answer: (a) \( 60(5!) \)

Question. Four couples (husband and wife) decide to form a committee of four members. The number of different committees that can be formed in which no couple finds a place is
(a) 10
(b) 12
(c) 14
(d) 16
Answer: (d) 16

Question. From 4 gentlemen and 6 ladies a committee of five is to be selected. The number of ways in which the committee can be formed so that gentlemen are in majority is
(a) 66
(b) 156
(c) 60
(d) None of the options
Answer: (a) 66

Question. There are 20 questions in a question paper. If no two students solve the same combination of questions but solve equal number of questions then the maximum number of students who appeared in the examination is
(a) \( {}^{20}C_9 \)
(b) \( {}^{20}C_{11} \)
(c) \( {}^{20}C_{10} \)
(d) None of the options
Answer: (c) \( {}^{20}C_{10} \)

Question. Nine hundred distinct n-digit positive numbers are to be formed using only the digits 2, 5 and 7. The smallest value of n for which this is possible is
(a) 6
(b) 7
(c) 8
(d) 9
Answer: (b) 7

Question. The total number of integral solutions for \( (x, y, z) \) such that \( xyz = 24 \) is
(a) 36
(b) 90
(c) 120
(d) None of the options
Answer: (c) 120

Question. The number of ways in which the letters of the word ARTICLE can be rearranged so that the even places are always occupied by consonants is
(a) 576
(b) \( {}^4C_3 \times (4!) \)
(c) \( 2(4!) \)
(d) None of the options
Answer: (a) 576

Choose the correct options. One or more options may be coorect.

Question. There are 10 bags \( B_1, B_2, B_3, \dots, B_{10} \), which contain 21, 22, \dots, 30 different articles respectively. The total number of ways to bring out 10 articles from a bag is
(a) \( {}^{31}C_{20} - {}^{21}C_{10} \)
(b) \( {}^{31}C_{21} \)
(c) \( {}^{31}C_{20} \)
(d) None of the options
Answer: (a) \( {}^{31}C_{20} - {}^{21}C_{10} \)

Question. If the number of arrangements of \( n - 1 \) things taken from n different things is k times the number of arrangements of \( n - 1 \) thing taken from n things in which two things are identical then value of k is
(a) \( \frac{1}{2} \)
(b) 2
(c) 4
(d) None of the options
Answer: (b) 2

Question. Kanchan has 10 friends among whom two are married to each other. She wishes to invite 5 of them for a party. If the married couple refuse to attend separately then the number of different ways in which she can invite friends is
(a) \( {}^8C_5 \)
(b) \( 2 \times {}^8C_3 \)
(c) \( {}^{10}C_5 - 2 \times {}^8C_4 \)
(d) None of the options
Answer: (b) \( 2 \times {}^8C_3 \) and (c) \( {}^{10}C_5 - 2 \times {}^8C_4 \)

Question. In a plane there are two families of lines \( y = x + r, y = -x + r \), where \( r \in \{0, 1, 2, 3, 4\} \). The number of squares of diagonals of the length 2 formed by the lines is
(a) 9
(b) 16
(c) 25
(d) None of the options
Answer: (a) 9

Question. There are n seats round a table numbered 1, 2, 3, \dots, n. The number of ways in which \( m (\leq n) \) persons can take seats is
(a) \( {}^nP_m \)
(b) \( {}^nC_m \times (m - 1)! \)
(c) \( {}^{n-1}P_{m-1} \)
(d) \( {}^nC_m \times m! \)
Answer: (a) \( {}^{nP}_m \) and (d) \( {}^nC_m \times m! \)

Question. Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \) and let \( \vec{r} \) be a variable vector such that \( \vec{r} \cdot \hat{i}, \vec{r} \cdot \hat{j} \) and \( \vec{r} \cdot \hat{k} \) are positive integers. If \( \vec{r} \cdot \vec{a} \leq 12 \) then the number of values of \( \vec{r} \) is
(a) \( {}^{12}C_9 - 1 \)
(b) \( {}^{12}C_3 \)
(c) \( {}^{12}C_9 \)
(d) None of the options
Answer: (b) \( {}^{12}C_3 \) and (c) \( {}^{12}C_9 \)

Question. The total number of ways in which a beggar can be given at least one rupee from four 25-paisa coins, three 50-paisa coins and 2 one-rupee coins, is
(a) 54
(b) 53
(c) 51
(d) None of the options
Answer: (a) 54

Question. For the equation \( x + y + z + w = 19 \), the number of positive integral solutions is equal to
(a) the number of ways in which 15 identical things can be distributed among 4 persons
(b) the number of ways in which 19 identical things can be distributed among 4 persons
(c) coefficient of \( x^{19} \) in \( (x^0 + x^1 + x^2 + \dots + x^{19})^4 \)
(d) coefficient of \( x^{19} \) in \( (x + x^2 + x^3 + \dots + x^{19})^4 \)
Answer: (a) the number of ways in which 15 identical things can be distributed among 4 persons and (d) coefficient of \( x^{19} \) in \( (x + x^2 + x^3 + \dots + x^{19})^4 \)