1.How many distinct arrangements of the letters in the word 'MATHEMATICS' can be formed if all the vowels are kept together?
2.Using the letters from the word 'LOGARITHMS', how many distinct 4-letter arrangements can be created without repeating any letter?
3.How many different groups consisting of 5 men and 2 women can be formed from a pool of 7 men and 3 women?
4.How many distinct ways can four books labeled A, B, C, and D be stacked vertically so that books A and B are never placed next to each other?
5.Given two identical red balls, two identical black balls, and two identical white balls, in how many distinct ways can these balls be arranged in the cells shown above so that no two balls of the same color are placed next to each other?
6.A club has twelve members, and each month, one member is randomly selected to host a dinner for the group. How many dinners will a specific member be expected to host in a year?
7.An individual ordered 5 pairs of black socks and an unknown number of brown sock pairs. Each black pair costs three times as much as a brown pair. However, when preparing the bill, the clerk mistakenly swapped the quantities of black and brown socks, resulting in the total bill doubling. How many pairs of brown socks were originally ordered?
8.A committee of two members, consisting of one male and one female, is to be formed from a group of five men and three women. Among the women, Ms. A refuses to serve on the committee if Mr. B is included. How many different committees can be formed under these conditions?
9.From six candidates interviewed, only two qualify for the principal position, while all six are eligible for the vice-principal roles. If one principal and two vice-principals need to be chosen, how many different selection combinations are possible?
10.A student must select 2 subjects from the following 5 options: Commerce, Economics, Statistics, Mathematics 1, and Mathematics 2. Note that Mathematics 2 can only be chosen if Mathematics 1 is also selected. How many different pairs of subjects can the student choose?
11.Three passengers—two men and one woman—are to be seated on a bus with five available seats, one of which is specifically reserved for females. The woman may choose to sit in the reserved seat or any other seat. How many distinct seating arrangements are possible for these three individuals?
12.Three standard six-sided dice are rolled simultaneously. How many possible outcomes are there where at least one of the dice displays the number 2?
13.In a mixed doubles tennis tournament, two teams will compete, each consisting of one male and one female player. There are four married couples available, but no team should include both husband and wife. What is the highest number of games that can be organized under these conditions?
14.A test consists of four multiple-choice questions, each having five possible answers with only one correct option per question. How many different answer combinations will result in the candidate not answering all four questions correctly?
15.How many distinct arrangements are possible for six players standing in a row if Asim and Raheem must not be positioned next to each other?
16.From five boys named A, B, C, D, and E, groups of three are to be formed such that no group includes both C and D simultaneously. What is the greatest number of such distinct groups possible?
17.A test consists of 10 true/false questions. Every candidate answers all questions, and no two candidates have the exact same sequence of answers. How many unique answer patterns are possible?
18.There are six points placed on one straight line and five points placed on another line parallel to the first. How many distinct straight lines can be drawn using all these points, including the two original lines?
19.From a pool of 12 seniors and 10 juniors, a committee of 10 members is to be formed. How many different ways can this committee be chosen if it must include at least one senior?
20.From a pool of 12 seniors and 10 juniors, a team of 10 members is to be formed. How many distinct ways can the team be chosen if it must consist of exactly 5 seniors and 5 juniors?