### Question. 1

How many integers in the set {100, 101, 102, ..., 999} have at least one digit repeated?

How many integers in the set {100, 101, 102, ..., 999} have at least one digit repeated?

In a tournament, there are 43 junior level and 51 senior level participants. Each pair of juniors play one match. Each pair of seniors play one match. There is no junior versus senior match. The number of girl versus girl matches in junior level is 153, while the number of boy versus boy matches in senior level is 276. The number of matches a boy plays against a girl is

How many four digit numbers, which are divisible by 6, can be formed using the digits 0, 2, 3, 4, 6, such that no digit is used more than once and 0 does not occur in the left-most position?

In how many ways can 8 identical pens be distributed among Amal, Bimal, and Kamal so that Amal gets at least 1 pen, Bimal gets at least 2 pens, and Kamal gets at least 3 pens?

In how many ways can 7 identical erasers be distributed among 4 kids in such a way that each kid gets at least one eraser but nobody gets more than 3 erasers?

Let AB, CD, EF, GH, and JK be five diameters of a circle with center at O. In how many ways can three points be chosen out of A, B, C, D, E, F, G, H, J, K, and O so as to form a triangle?

The number of solutions (x, y, z) to the equation x – y – z = 25, where x, y, and z are positive integers such that x ≤ 40, y ≤ 12, and z ≤ 12 is

Arun's present age in years is 40% of Barun's. In another few years, Arun's age will be half of Barun's. By what percentage will Barun's age increase during this period?

The first n natural numbers, 1 to n, have to be arranged in a row from left to right. The n numbers are arranged such that there are an odd number of numbers between any two even numbers as well as between any two odd numbers. If the number of ways in which this can be done is 72, then find the value of n

If we arrange the letters of the word ‘KAKA’ in all possible ways, what is the probability that vowels will not be together in an arrangement?

Three persons - A, B and C - are playing the game of death. 3 bullets are placed randomly in a revolver having 6 chambers. Each one has to shoot himself by pulling the trigger once after which the revolver passes to the next person. This process continues till two of them are dead and the survivor of the game becomes the winner. What is the probability that B is the winner if A starts the game and A, B and C take turns in that order.

Amar, Akbar and Antony are three students in a class of 9 students. A class photo is taken. The number of ways in which it can be taken such that no two of Amar, Akbar and Antony are sitting together is:

There are exactly sixty chairs around a circular table. There are some people sitting on these chairs in such a way that the next person to be seated around the table will have to sit next to someone. What is the least possible number of people sitting around the table currently?

In how many ways can 18 identical candies be distributed among 8 children such that the number of candies received by each child is a prime number?

What is the probability that the product of two integers chosen at random has the same unit digit as the two integers?

A box contains five yellow and five green balls. A ball is picked from the box and is replaced by a ball of the other colour. For instance, if a green ball is picked then it is replaced by a yellow ball and vice-versa. The process is repeated ten times and then a ball is picked from the box. What is the probability that this ball is yellow?

A cube is painted with red colour and then cut into 64 small identical cubes. If two cubes are picked randomly from the heap of 64 cubes, what is the probability that both of them have exactly two faces painted red?

In how many ways can 6 letters A, B, C, D, E and F be arranged in a row such that D is always somewhere between A and B?

A and B throw with one dice for a stake of Rs. 11 which is to be won by the player who first throws 6. If A has the first throw, what are their respective expectations

A man, while driving to his office, finds three traffic signals on his way. The probability that the traffic light is red when he reaches the first, second and third traffic signal is 5/8, 5/6 and 3/5 respectively. What is the probability that he finds at least one traffic light on his way which is not red?

**Directions for Questions: **The figure below shows the plan of a town. The streets are at right angles to each other. Arectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.

Neelam rides her bicycle from her house at A to her club at C, via B taking the shortest path. Then the number of possible shortestpaths that she can choose is

Neelam rides her bicycle from her house at A to her office at B, taking the shortest path. Then the number of possible shortestpaths that she can choose is

Suppose you have a currency, named Miso, in three denominations: 1 Miso, 10 Misos and 50 Misos. In how many ways can youpay a bill of 107 Misos?

**Directions for Questions:** Let S be the set of all pairs (i, j) where 1 ≤ i< j ≤ n and n ≥ 4. Any two distinct members of S arecalled “friends” if they have one constituent of the pairs in common and “enemies” otherwise. For example, if n = 4, then S = {(1, 2),(1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}. Here, (1, 2) and (1, 3) are friends, (1, 2) and (2, 3) are also friends, but (1, 4) and (2, 3) are enemies.

For general n, consider any two members of S that are friends. How many other members of S will be common friends of both thesemembers?

For general n, how many enemies will each member of S have?

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

Let S be a set of positive integers such that every element n of S satisfies the conditions

1. 1000 ≤ n ≤ 1200

2. every digit of n is odd

Then how many elements of S are divisible by 3?

Let n! = 1 × 2 × 3 × ..... × n for integer n≥1. If p = 1! + (2 × 2!) + (3 × 3!) + ..... + (10 × 10!), then p + 2 when divided by 11! leavesa remainder of

Let S be the set of five-digit numbers formed by the digits 1, 2, 3, 4 and 5, using each digit exactly once such that exactly two oddpositions are occupied by odd digits. What is the sum of the digits in the rightmost position of the numbers in S?

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

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

A new flag is to be designed with six vertical stripes using some or all of the colours yellow, green, blue and red. Then, the numberof ways this can be done such that no two adjacent stripes have the same colour out is

In the adjoining figure, the lines represent one-way roads allowing travel only northwards or only westwards. Along how manydistinct routes can a car reach point B from point A?

Each family in a locality has at most two adults, and no family has fewer than 3 children. Considering all the families together, thereare more adults than boys, more boys than girls, and more girls than families. Then the minimum possible number of families inthe locality is

Suppose n is an integer such that the sum of the digits of n is 2, and 10^{10 }< n < 10^{11}. The number of different values for n is

N persons stand on the circumference of a circle at distinct points. Each possible pair of persons, not standing next to each other,sings a two-minute song one pair after the other. If the total time taken for singing is 28 minutes, what is N?

An intelligence agency forms a code of two distinct digits selected from 0, 1, 2, ............ , 9 such that the first digit of the code isnonzero. The code, handwritten on a slip, can however potentially create confusion when read upside down –– for example, thecode 91 may appear as 16. How many codes are there for which no such confusion can arise?

There are 12 towns grouped into four zones with three towns per zone. It is intended to connect the towns with telephone linessuch that every two towns are connected with three direct lines if they belong to the same zone, and with only one direct lineotherwise. How many direct telephone lines are required?

**Directions for questions: Read the information given below and answer the questions that follow :**

A string of three English letters is formed as per the following rules :

(a) The first letter is any vowel.

(b) The second letter is m, n or p

(c) If the second letter is m then the third letter is any vowel which is different from the first letter

(d) If the second letter is n then the third letter is e or u.

(e) If second letter is p then the third letter is the same as the first letter

How many strings of letters can possibly be formed using the above rules such that the third letter of the string is e?

How many strings of letters can possibly be formed using the above rules?

A graph may be defined as a set of points connected by lines called edges. Every edge connects a pair of points. Thus, a triangleis 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 agraph with three points of degree 2 each. Consider a graph with 12 points. It is possible to reach any point from any other pointthrough a sequence of edges. The number of edges, e, in the graph must satisfy the condition

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 boxcontains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which thiscan be done is

How many three digit positive integers, with digits x, y and z in the hundred’s, ten’s and unit’s place respectively, exist such that x<y, z< y and x≠0?

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

If there are 10 positive real numbers n_{1} < n_{2} < n_{3} ...... < n10. How many triplets of these numbers (n_{1}, n_{2}, n_{3}), (n_{2}, n_{3}, n_{4}), ..... canbe generated such that in each triplet the first number is always less than the second number and the second number is alwaysless than the third number?

If there are 10 positive real numbers n1 < n2 < n3 ... < n10. How many triplets of these numbers (n1, n2, n3), (n2, n3, n4), ... can begenerated such that in each triplet the first number is always less than the second number, and the second number is always lessthan the third number?

How many numbers greater than 0 and less than a million can be formed with the digits of 0, 7 and 8?

In how many ways is it possible to choose a white square and a black square on a chess board so that the squares must not liein the same row or column?

**Directions for questions: Read the information given below and answer the questions that follow :**

Each of the 11 letters A, H, I, M, O, T, U, V, W, X and Z appears same when looked at in a mirror. They are called symmetric letters.Other letters in the alphabet are asymmetric letters.

How many three-letter computer passwords can be formed (no repetition allowed) with at least one symmetric letter?

How many four-letter computer passwords can be formed using only the symmetric letters (no repetition allowed)?

10 straight lines, no two of which are parallel and no three of which pass through any common point, are drawn on a plane. Thetotal number of regions (including finite and infinite regions) into which the plane would be divided by the lines is

Let n be the number of different 5 digit numbers, divisible by 4 with the digits 1, 2, 3, 4, 5, and 6, no digit being repeated in thenumbers. What is the value of n?

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

Ashish is given Rs. 158 in one rupee denominations. He has been asked to allocate them into a number of bags such that anyamount required between Re. 1 and Rs. 158 can be given by handing out a certain number of bags without opening them. Whatis the minimum number of bags required?

A red light flashes 3 times per minute and a green light flashes 5 times in two minutes at regular intervals. If both lights startflashing at the time, how many times do they flash together in each hour?

There are five boxes each of a different weight and none weighing more than 100. Arun weights two boxes at a time and obtainsthe following readings in grams : 110, 112, 113, 114, 116, 117, 118, 119, 120, 121. What is the weight of the heaviest box?

**Directions for questions: Read the information given below and answer the questions that follow :**

The tournament for ABC Cup is arranged as per the following rules: in the beginning 16 teams are entered and divided in 2 groupsof 8 teams each where the team in any group plays exactly once with all the teams in the same group. At the end of this round topfour teams from each group advance to the next round in which two teams play each other and the losing team goes out of thetournament. The rules of the tournament are such that every match can result only in a win or a loss and not in a tie. The winnerin the first round takes one point from the win and the loser gets zero. In case of tie on a position the rules are very complex andinclude a series of deciding measures.

Which of the following statements about a team which has already qualified for the second round is true?

The minimum number of matches that a team must win in order to qualify for the second round is

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

What is the total number of matches played in the tournament?

One red, three white and two blue flags are to be arranged in such a way that no two flags of the same colour are adjacent and theflags at the two ends are of different colours. The number of ways in which this can be done is

X is an odd integer such that 100 < x < 200 and x is divisible by 3 but not 7. The possible number of values of x is

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 the 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?

Sameer has to make a telephone call to his friend Harish Unfortunately he does not remember the 7- digit phone number. But he remembers that the first 3 digits are 635 or 674, the number is odd and there is exactly one 9 in the number. The minimum number of trials that Sameer has to make to be successful is

Out of 2n+1 students, n students have to be given the scholarships. The number of ways in which at least one student can begiven the scholarship is 63. What is the number of students receiving the scholarship?

There are three boxes with 2 red, 2 white and one red and one white ball. All of them are mislabelled. You have to correct all the labels by picking only one ball from any one of the boxes. Which is the box that you would like to open

There are 10 points on a line and 11 points on another line, which are parallel to each other. How many triangles can be drawn taking the vertices on any of the line?

Five persons A, B, C, D and E along with their wives are seated around a round table such that no two men are adjacent to each other. The wives are three places away from their husbands. Mrs. C is on the left of Mr. A, Mrs. E is two places to the right of Mrs. B. Then, who is on the right hand side of Mr. A?

A, B, C, D, ..................X, Y, Z are the players who participated in a tournament. Everyone played with every other player exactly once. A win scores 2 points, a draw scores 1 point and a loss scores 0 points. None of the matches ended in a draw. No two players scored the same score. At the end of the tournament, the ranking list is published which is in accordance with the alphabetical order. Then

How many five digit numbers can be formed using 2,3,8,7,5 exactly once such that the number is divisible by 125?

How many numbers can be formed from 1,2,3,4 and 5 (without repetition), when the digit at the units place must be greater than that in the tenth place?

ABC is a three-digit number in which A > 0. The value of ABC is equal to the sum of the factorials of its three digits. What is the value of B?

In how many ways can the eight 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?

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?

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?

A,B,C and D are four towns any three of which are non-colinear. 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

I. The probability of encountering 54 Sundays in a leap year

II. The probability of encountering 53 Sundays in a non-leap year