Three employee-mentor pairs from different departments are travelling to an offsite workshop. The six people will occupy the second and third rows of a cab, with each row having exactly three seats. To avoid distractions, an employee and their corresponding mentor cannot sit next to each other in the same row, and no employee may sit directly in front of or behind their corresponding mentor.

How many seating arrangements are possible?

A stationery shop has tokens available in \(3\) different colours (at least four tokens of each colour). Tokens are identical except for their colour. Let \(m\) denote the number of arrangements of \(4\) tokens such that not all four tokens are of the same colour. Let \(n\) denote the number of arrangements of \(4\) tokens such that every arrangement contains at least one token of each colour.

Which of the following relations is correct?

Among all four-digit positive integers greater than \(4000\), how many have exactly three identical digits and one digit that is different from the other three?

During a peer-learning activity, study cards numbered \(1, 2, 3, 4, 5, 6, 7,\) and \(8\) are available. A group must contain three or more distinct cards. Every group must include cards \(3\) and \(5\), but cards \(7\) and \(8\) can never be included together.

How many such groups can be formed?

For a college alumni meet in Mumbai, a group of \(10\) people consists of \(3\) married couples and \(4\) single men. A committee of \(4\) people has to be formed from these \(10\) people. How many different committees can be formed if the committee can consist of at most \(1\) married couple?

For a digital practice portal, a four-digit positive integer has to be created using digits from \(1\) to \(9\). The integer must be such that two digits are equal to each other, and the remaining two digits are also equal to each other but different from the first pair.

How many such four-digit positive integers can be formed?

For a login system, a \(4\)-digit passcode has to be formed using the digits \(0\) to \(9\). The passcode must be a valid \(4\)-digit number and must contain exactly \(3\) distinct digits.

How many such \(4\)-digit numbers can be formed?

A library assistant at a Delhi coaching centre has a set of \(26\) reference volumes, one for each letter of the alphabet. There is a special shelf with \(26\) slots in a row, and each slot is labelled alphabetically. The assistant has to place every volume back into its correct labelled slot. During the process, there must never be a gap between any of the volumes already placed on the shelf. This means that after the first volume is placed, every next volume must be placed adjacent to at least one volume already on the shelf.

If the assistant can start with any of the \(26\) volumes, in how many ways can the task be completed?

At a Bengaluru coaching centre, \(3\) different prizes are to be distributed among \(5\) students. Each prize must be given to one student, but no student should get all the prizes.

In how many ways can the prizes be distributed?

In a number puzzle, students are asked to count positive integers less than \(10,000\). How many such positive integers are there in which the sum of the digits is equal to \(5\)?

At a college fest in Pune, coupons of values \(1\), \(5\), \(10\), and \(25\) points are available. In how many different ways can these coupons be combined to make a total of \(110\) points, if at least one coupon of each type must be included?

For a college fest in Jaipur, a committee of \(3\) students has to be formed from \(5\) candidates: Ananya, Bhavna, Pranav, Sarthak, and Kavya. Pranav and Sarthak refuse to be on the committee together, and Ananya refuses to be on the committee without Pranav.

How many different committees can be formed?

In a chess tournament at a college fest, \(3\) women and a few men participated. Each player played two matches with each of the other players. The number of matches that men played among themselves was \(78\) more than the number of matches they played with the women.

How many more men than women participated in the tournament?

A college sports league has four zones. The four zones have \(9, 10, 11,\) and \(12\) teams respectively qualifying for the playoffs. Each zone conducts its own double-elimination tournament, where a team is eliminated only after losing \(2\) games. The four zone champions then play a single-elimination tournament, where a team is eliminated after losing \(1\) game, to decide the overall champion.

Assuming there are no ties and no forfeits, what is the maximum number of games that could be played to decide the overall champion?

A app generates seven-digit positive integer IDs. How many such seven-digit IDs include the sequence \(123\) in that order? For example, \(1234567\) and \(9991239\) are valid IDs.

For a college event, \(P_n\) denotes the number of ways of selecting \(3\) volunteers from \(n\) people sitting in a straight row, such that no two selected people are sitting next to each other. Similarly, \(Q_n\) denotes the number of ways of selecting \(3\) volunteers from \(n\) people sitting around a circular table, such that no two selected people are sitting next to each other. If \(P_n - Q_n = 6\), then what is the value of \(n\)?

At a student convention in Bengaluru, there were \(9\) sets of twins and \(6\) sets of triplets, all from different families. Each twin shook hands with all the twins except his or her sibling, and with half the triplets. Each triplet shook hands with all the triplets except his or her siblings, and with half the twins.

How many handshakes took place?

For a test portal, a database password is made only of digits, and no digit can be repeated. The password is known to contain at least \(8\) digits. If it takes \(12\) seconds to try one password combination, what is the amount of time, in minutes, necessary to guarantee access to the database?

At a school photography session, \(6\) students of \(6\) different heights have to be arranged in two rows of \(3\) students each. Each student in the first row stands directly in front of one student in the second row. Within each row, heights must increase from left to right. Also, each student in the second row must be taller than the student standing directly in front of him or her.

How many such arrangements are possible?

In a board-game club at an MBA college, a bag contains five \(6\)-sided dice numbered \(1\) to \(6\), three \(12\)-sided dice numbered \(1\) to \(12\), and two \(20\)-sided dice numbered \(1\) to \(20\). Four dice are selected from the bag and then rolled. How many different possible totals can be obtained as the sum of the numbers showing on the four dice?

An English-medium school and a Hindi-medium school are both managed by the same education trust. There is \(1\) vacant superintendent post, \(4\) vacant teaching posts in the English-medium school, and \(4\) vacant teaching posts in the Hindi-medium school. There are altogether \(11\) candidates for these appointments, of whom \(3\) apply exclusively for the superintendent post and \(2\) apply exclusively for the English-medium school teaching posts.

In how many ways can these different appointments be made?

At a puzzle session, the \(8\) vertices of a cube are to be labelled using the integers \(1\) through \(8\), with each integer used exactly once. The labels must be placed so that the sum of the four labels on every face of the cube is the same. Arrangements that can be obtained from one another by rotating the cube are considered the same.

How many different arrangements are possible?

In a geometry activity, there are \(m\) marked points on the straight line \(AB\) and \(n\) marked points on the straight line \(AC\). None of these marked points is point \(A\), and points \(A\), \(B\), and \(C\) are not collinear. Triangles are formed using these marked points as vertices in two cases: (i) Point \(A\) is excluded. (ii) Point \(A\) is included.

What is the ratio of the number of triangles in case (i) to the number of triangles in case (ii)?

A student has forgotten his \(6\)-digit college portal ID. He remembers that the first two digits are either \(15\) or \(26\), the ID is even, and the digit \(6\) appears exactly twice.

If he uses trial and error to find the ID, at most how many trials will he need to succeed?

At a college fest stall, a seller has \(5\) chocolates each of \(3\) different varieties. He has to sell \(9\) chocolates to \(9\) different people. He can sell chocolates of at most \(2\) varieties.

In how many different ways can he sell the chocolates?