Practice Test


1) The number of three digit numbers having only two consecutive digits identical is


2) The total number of seven-digit numbers the sum of whose digits is even is


3) The total number of numbers of not more than 20 digits that are formed by using the digits 0, 1, 2, 3 and 4 is


4) The number of six digit numbers that can be formed from the digits 1,2,3,4,5,6 and 7 so that digits do not repeat and the terminal digits are even is


5) Three dice are rolled. The number of possible outcomes in which at least one die shows 5 is


6) The number of ways in which n distinct objects can be put into two different boxes is


7) The total number of all proper factors of 75600 is


8) There are 6 tasks and 6 persons. Task 1 cannot be assigned either to person 1 or to person 2; task 2 must be assigned to either person 3 or person 4. Every person is to be assigned one task. In how many ways can the assignment be done ?


9) The number of ways in which one or more balls can be selected out of 10 white, 9 green and 7 blue balls is


10) If 12 persons are seated in a row, the number of ways of selecting 3 persons from them, so that no two of them are seated next to each other is


11) The number of all possible selections of one or more questions from 10 given questions, each question having one alternative is


12) A lady gives a dinner party to 5 guests to be selected from nine friends.The number of ways of forming the party of 5, given that two of the friends will not attend the party together is


13) All possible two factors products are formed from the numbers 1,2,3,4,...,200. The number of factors out of total obtained which are multiples of 5 is


14) the sides AB, BC, CA of a triangle ABC have 3,4 and 5 interior points respectively on them. The total number of triangles that can be constructed by using these points as vertices is


15) The different letters of an alphabet are given, words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated is


16) A five digit number divisible by 3 is to be formed using the numerals 0,1,2,3,4 and 5, without repetition. The total number of ways this can be done is


17) The number of possible outcomes in a throw of an ordinary dice in which at least one of the dice shows an odd number is


18) If all permutations of the letters of the word AGAIN are arranged as in dictionary, then fiftieth word is


19) In a chess tournament, where the participants were to play one game with another, two chess players fell ill, having played 3 games each. If the total number of games played is 84, the number of participants at the beginning was


20) 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 atleast one black ball is to be included in the draw ?


21) The number of ways in which a mixed doubles game in tennis can be arranged from 5 married couples, if no husband and wife play in the same game, is


22) Six teachers and six students have to sit around a circular table such that there is a teacher between any two students. The number of ways in which they can sit is


23) The number of ways in which 13 gold coins can be distributed among three persons such that each one gets at least two gold coin is :


24) The number of seven digit numbers divisible by 9 formed with digits 1,2,3,4,5,6,7,8,9 without repetition is


25) All the words that can be formed using alphabets A, H, L, U and R are written as in a dictionary ( no alphabet is repeated.) Rank of the word RAHUL is


26) The number of six digit numbers that can be formed from the digits 1,2,3,4,5,6 and 7 so that digits do not repeat and the terminal digits are even, is


27) The number of ways in which four letters of the word MATHEMATICS can be arranged is given by


28) ABCD is a convex quadrilateral. 3,4,5 and 6 points are marked on the sides AB, BC, CD and DA respectively. The number of triangles with vertices on different sides is


29) In a polygon on 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 the polygon is


30) m distinct animals of a circus have to be placed in m cages, one in each cage. If n ( < m ) cages are too small to accommodate p ( n < p < m ) animals, then the number of ways of putting the animals into cages are


31) Total number of four digit odd numbers that can be formed using 0,1,2,3,5,7 ( using repetition allowed ) are


32) Number greater than 1000 but less than 4000 is formed using the digits 0,1,2,3,4 ( repetition allowed) is


33) Five digit number divisible by 3 is formed using 0,1,2,3,4,6 and 7 without repetition. Total number of such numbers are


34) The sum of integers from 1 to 100 that are divisible by 2 or 5 is


35) The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by


36) How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions ?


37) If two dices are tossed simultaneously, the number of elements in the resulting sample space is :


38) From amongst 36 teachers in a school, one principal and one vice-principal are to be appointed. In how many ways can this be done ?


39) A boy has 3 library cards and 8 books of his interest in the library.Of these 8, he does not want to borrow chemistry part II unless Chemistry part I is also borrowed. In how many ways can he choose the three books to be borrowed ?


40) In how many ways can six different rings be worn on four fingers of one hand ?


41) There are three prizes to be distributed among five students. If no student gets more than one prize, then this can be done in :


42) There are 6 equally spaced points A,B,C,D,E and F marked on a circle with radius R. How many convex pentagons of distinctly different areas can be drawn using these points as vertices ?


43) Ten different letters of an alphabet are given. Words with 5 letters are formed from these given letters. then the number of words which have at least one letter repeated is :


44) In a hockey championship there were 153 matches played. Every two teams played one match with each other. The number of teams participating in the championship is :


45) Find the number of ways in which 8064 can be resolved as the product of two factors ?


46) In an examination paper there are two sections each containing 4 questions. A candidate is required to attempt 5 questions but not more than 3 questions from any particular section. In how many ways can 5 questions be selected ?


47) A box contains 10 balls but out of which 3 are red and the rest are blue. In how many ways can a random sample of 6 balls be drawn from the bag so that at the most 2 red balls are included in the sample and no sample has all the 6 balls of same colour ?


48) Three boys and three girls are to be seated around a table in a circle. Among them the boy X does not want any girl neighbour and the girl Y does not want any boy neighbour. How many such arrangements are possible ?


49) Two series of a question booklets for an aptitude test are to be given to twelve students. In how many ways can the students be placed in two rows of six each so that there should be no identical series side by side and that the students sitting one behind the other should have the same series ?


50) A class photograph has to be taken. The front row consists of 6 girls who are sitting. 20 boys are standing behind. The two corner positions are reserved for 2 tallest boys. In how many ways can the students be arranged ?


51) In how many ways can twelve girls be arranged in a row if two particular girls must occupy the end places ?


52) In how many ways can a selection of 5 letters be made out of 5 A's, 4B's, 3 C's, 2 D's and 1 E's ?


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


54) Boxes numbered 1,2,3,4 and 5 are kept in a row and they are to be filled with either a red or a blue ball, such that no two adjacent boxes can be filled with blue balls. Then how many different arrangements are possible, given that all balls of a given colour are exactly identical in all respects ?


55) A man has nine friends - four boys and five girls. In how many ways can he invite them, if there have to be exactly three girls in the invitees ?


56) In how many ways can the directors, the Vice-chairman and the Chairman of a firm be seated at a round-table, if the Chairman has to sit between the Vice-chairman and the Director ?


57) Out of 2n +1 students, n students have to be given the scholarships. The number of ways in which at least one student can be given the scholarship is 6.3. What is the number of students receiving the scholarship ?


58) There are three books on table A which has to be moved to table B. The order of the book on table A was 1,2,3, with book 1 at the bottom. The order of the book on table B should be with book 2 on top and book 1 on bottom. Note that you can pick up the books in order they have been arranged. You can't remove the books from the middle of the stack. In how many minimum steps can we place the books on table B in the required order ?


59) The maximum number of matches that a team going out of the tournament in the first round itself can win is


60) Twenty seven persons attend a party. Which one of the following statements can never be true ?


61) There are 6 boxes numbered 1,2,.....6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is :


62) A graph may be defined as a set of points connected by lines called edges. Every edge connects a pair of points. Thus, a triangle is a graph with 3 edges and 3 points. The degree of a point is the number of edges connected to it. For example, a triangle is a graph with three points of degree 2 each. Consider a graph with 12 points. It is possible to reach any point from any other point through a sequence of edges. The number of edges, e, in the graph must satisfy the condition.


63) In a chess competition involving some boys and girls of a school, every student had to play exactly one game with every other students. It was found that in 45 games both the players were girls and in 190 games both were boys. The number of games in which one player was a boy and the other was a girl is


64) Three Englishmen and three Frenchmen work for the same company. Each of them knows a secret not known to others. They need to exchange these secrets over person-to-person phone calls so that eventually each person knows all six secrets. None of the Frenchmen knows English, and only one Englishmen knows French. What is the minimum number of phone calls needed for the above purpose ?


65) To fill a number of vacancies, an employer must hire 3 programmers from among 6 applicants, and 2 managers from among 4 applicants. What is the total number of ways in which she can make her selection ?


66) There are 100 students in a particular class. 60% students play cricket, 30 % student play football and 10 % students play both the games. What is the number of students who play neither cricket nor football ?


67) There are four routes to travel from city A to city B and six routes from city B to city C. How man routes are possible to travel from the city A to city B ?


68) In a given race the odds in favour of three horses A, B, C are 1:3; 1:4; 1:5 respectively. Assuming that dead head is impossible the probability that one of them wins is


69) A man and his wife appear for an interview for two posts. The probability of the husband's selection is 1/7 and that of the wife's selection is 1/5. The probability that only one of them will be selected is


70) If a leap year selected at random, the chance that it will contain 53 Sunday is


71) A positive integer N is selected such that 100 < N < 200. The probability that it is divisible by either 4 or 7 is :


72) In a single throw with four dice, the probability of throwing seven is


73) If three vertices of a regular hexagon are chosen at random, then the chance that they form an equilateral triangle is :


74) Six dices are thrown. The probability that different number will turn up is :


75) Four balls are drawn at random from a bag containing 5 white, 4 green and 3 black balls. The probability that exactly two of them are white is :


76) There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, in a random order till both the faulty machines are identified. Then the probability that only two tests are needed is


77) The probability that a person will hit a target in shooting practice is 0.3. If he shoots 10 times, the probability that he hits the target is


78) The probability that A can solve a problem is 2/3 and B can solve it is 3/4. If both attempt the problem, what is the probability that the problem get solved ?


79) If the probability that A and B will die within a year are p and q respectively, then probability that only one of them will be alive at the end of the year is


80) Three integers are chosen at random from the first 20 integers. The probability that their product is even, is


81) A die is loaded in such a way that each odd number is twice al likely to occur as each even number. If E is the event of a number greater than or equal to 4 on a single toss of the die, then P(E) is :


82) The probability that two integers chosen at random and their product will have the same last digit is :


83) Seven people seat themselves indiscriminately at round table. The probability that two distinguished persons will be next to each other is


84) If two squares are chosen at random on a chess board, the probability that they have a side in common is


85) A speaks the truth in 70 percent cases and B in 80 percent. The probability that they will contradict eact. other when describing a single event is


86) One ticket is selected at random from 100 tickets numbered 00, 01, 02, .... 99. Suppose S and T are the sum and product of the digits of the number on the ticket, then P ( S = 9 / T = 0) is


87) A student appears for tests I, II and III. The student is successful if he passes either in tests I and II or tests I and III. The probability of the student passing in tests I, II, III are p,q and 1/2 respectively. The probability that the student is successful is 1/2 then the relation between p and q is given by


88) A die is loaded such that the probability of throwing the number i is proportional to its reciprocal. The probability that 3 appears in a single throw is :


89) The probability of getting 10 in a single throw of three fair dice is :


90) The probability that when 12 balls are distributed among three boxes, the first will contain three balls is,


91) A and B toss a fair coin each simultaneously 50 times. The probability that both of them will not get tail at the same toss is


92) If A and B are two independent events and P(C) = 0, then A, B, C are :


93) There is a five volume dictionary among 50 books arranged on a shelf in random order. If the volumes are not necessarily kept side by side, the probability that they occur in increasing order from left to right is :


94) If n integers taken at random are multiplied together, then the probability that the last digit of the product is 1,
3,7 or 9 is


95) A programmer noted the results of attempting to run 20 programs. The results showed that 2 programs ran correctly in the first attempt, 7 ran correctly in the second attempt, 5 ran correctly in the third attempt, 4 ran correctly in the fourth attempt and 2 ran correctly in the fifth attempt. What is the probability that his next programme will run correctly on the third run ?


96) Course materials are sent to students by a distance education institution. The probability that they will send a wrong programme's study material is 1/5. There is a probability of 3/4 that the package is damaged in transit, and there is a probability of 1/3 that there is a short shipment. What is the probability that the complete material for the course arrives without any damage in transit ?


97) A coin is tossed 5 times. What is the probability hat head appears an odd number of times ?


98) Atul can hit a target 3 times in 6 shots, Bhola can hit the target 2 times in 6 shots and Chandra can hit the 4 times in 4 shots. What is the probability that at least 2 shots ( out of 1 shot taken by each one of them ) hit the target ?


99) A bag contain 5 white, 7 red and 8 black balls. If 4 balls are drawn one by one with replacement, what is the probability that all are white ?


100) A dice is thrown 6 times. If 'getting an odd number' is a 'success', the probability of 5 successes is :


101) A bag has 4 red and 5 black balls. A second bag has 3 red and 7 black balls. One ball is drawn from the first bag and two from the second. The probability that there are two black balls and a red ball is :


102) Four different objects 1,2,3,4 are distributed at random in four places marked 1,2,3,4. What is the probability that none of the objects occupy the place corresponding to its number ?


103) There are 6 positive and 8 negative numbers. Four numbers are chosen at random and multiplied. The probability that the product is a positive number is :


104) Two dice are tossed. The probability hat the total score is a prime number is :


105) A bag contains 3 white balls and 2 black balls. Another bag contains 2 white balls and 4 black balls. A bag is taken and a ball is picked at random from it. The probability that the ball will be white is :


106) One hundred identical coins each with probability p showing up heads are tossed. If 0 < p < 1 and the probability of heads showing on 50 coins is equal to that of heads on 51 coins, then the value of p is :


107) Suppose six coins are tossed simultaneously. Then the probability of getting at least one tail is :


108) The probability that a student is not a swimmer is 1/5. Then the probability that out of five students, four are swimmers is :


109) A bag contains 2 red, 3 green and 2 blue balls. 2 balls are to be drawn randomly. What is probability that the balls drawn contain no blue ball ?


110) I forgot the last digit of a 7 digit telephone number. If I randomly dial the final 3 digits after correctly dialing the first four, then what is the chance of dialing the correct number ?


111) In his wardrobe, Timothy has 3 trousers. One of them is black and second blue, and the third brown. In the wardrobe, he also has 4 shirts. One of them is black and the other 3 are white. He opens his wardrobe in the dark and picks out one shirt-trouser pair without examining their colour. What is the liklihood that neither the shirt nor the trouser is black ?


112) A car is parked by an owner amongst 25 cars in a row, not at either end. On his return he finds that exactly 15 places are still occupied. The probability that both the neighbouring places are empty is


113) A fair coin is tossed repeatedly. If the tail appears on first four tosses, then the probability of the head appearing on the fifth toss equals


114) The probability that the birth days of six different persons will fall in exactly two calender months is


115) A problem in mathematics is given to three students A,B,C and their respective probability of solving the problem is 1/2,1/3 and 1/4. Probability that the problem is solved is


116) Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The probability that Mr. A selected the winning horse is


117) The probability that A speaks truth is 4/5, while this probability for B is 3/4. The probability that thy contradict each other when asked to speak on face is


118) The probabilities of four cricketers A,B, C and D scoring more than 50 runs in a match are 1/2, 1/3, 1/4 and 1/10. It is known that exactly two of the players scored more than 50 runs in a particular match. The probability that these players were A and B is


119) 2n boys are randomly divided into two subgroups containing n boys each. The probability that the two tallest boys are in different groups is


120) How many words are there to arrange the letters in the word GARDEN with vowels in alphabetical order