50 Counting Questions#

Formulas and Notation#

  • Premutations: \(P(n,k)=\frac{n!}{(n-k)!}\) , the number of ordered ways to permute \(n\) objects into \(k\) bins.

  • Combinations: \(\binom{n}{k}=\frac{n!}{k!(n-k)!}\) , the number of unordered ways to arrange \(n\) objects into \(k\) bins. This notation also refers to the binomial coefficients and is read ``\(n\) choose \(k\)’’.

  • Cardinality Operator: \(n(S)\) , counts the (possibly infinite) number elements in set \(S\).

  • We will use “word’’ to refer to all possible distinct orderings of a group of letters without concern for whether or not these orderings would appear in a dictionary.

Problems#

  1. There are \(16\) students in the Math club. How many different ways could they select a president, vice president and treasurer for the club?

  2. A website requires the user to choose a password with \(5\) letters and \(2\) numbers in that order. Each letter or digit may be used only once. How many different passwords are possible? How does this change if \(5\) letters and \(2\) numbers may be used in any order?

  3. The name of \(11\) students are placed in a hat. A teacher will reach into the hat and select \(3\) names at one time. Each of those \(3\) students will win the same prize. How many different groups of \(3\) winners could be chosen?

  4. In how many ways can a sorority of \(20\) members select a president, vice president and treasury, assuming that the same person cannot hold more than one office?

  5. Out of \(7\) consonants and \(4\) vowels, how many “words” of \(3\) consonants and \(2\) vowels can be formed?

  6. A license plate is to consist of three letters followed by two digits. How many different license plates are possible if the first letter must be a vowel (a, e, i, o, u), and repetition of letters is not permitted, but repetition of digits is permitted?

  7. A teacher is making a multiple-choice quiz. She wants to give each student the same questions, but have each student’s questions appear in a different order. If there are twenty-seven students in the class, what is the least number of questions the quiz must contain?

  8. How many \(10\) digit numbers can be formed using \(3\) and \(7\) only?

  9. Everyone shakes hands with everyone else in a room. Total number of handshakes is 66. What is the total number of people in the room?

  10. In a college football training session, the defensive coordinator needs to have 10 players standing in a row. Among these \(10\) players, there is a freshmen, \(2\) sophomores, \(4\) juniors, and \(3\) seniors, respectively. How many different ways can they be arranged in a row if only their class level will be distinguished?

  11. In one year, three awards (research, teaching, and service) will be given for a class of \(25\) graduate students in a statistics department. If each student can receive at most one award, how many possible selections are there?

  12. A president and a treasurer are to be chosen from a student club consisting of 50 people. How many different choices of officers are possible if the first chosen will be president?

  13. A developer of a new subdivision offers a prospective home buyer a choice of 4 designs, 3 different heating systems, a garage or carport, and a patio or screened porch. How many different plans are available to this buyer?

  14. In how many different ways can a true-false test consisting of 8 questions be answered?

  15. Find the number of ways to arrange \(16\) items in groups of \(4\) at a time (order matters).

  16. Find the number of ways to take \(20\) objects and arrange them in groups of \(5\) at a time where order does not matter?

  17. How many ways are there to select a subcommittee of \(7\) members from among a committee of \(17\)?

  18. How many ways are there to have a license plate with \(3\) letters followed by \(4\) numbers in which the \(4\) numbers are listed from least to greatest (examples: \(1234\) or \(0569\))?

  19. How many full house combinations are there in 5-card poker?

  20. From a group of \(7\) men and \(6\) women, five persons are to be selected to form a committee so that at least \(3\) men sit on the committee. In how many ways can it be done?

  21. In how many different ways can the letters of the word ‘LEADING’ be arranged in such a way that the vowels are together?

  22. Ten people go to a party. How many different ways can they be seated at a round table?

  23. There are \(10\) marbles in a bag numbered from \(1\) to \(10\). Three marbles are selected at random without replacement. How many different ways are there of selecting the three marbles?

  24. How many ways can I give 3 tin cans to 8 people if each person receives at most one can?

  25. An identification code is to consist of seven letters followed by three digits. How many different codes are possible if repetition is permitted?

  26. An 8-bit binary word is a sequence of 8 digits, of which each may be either a 0 or a 1. How many different 8-bit words are there?

  27. Jane and Thomas are among the 8 people from which a committee of 4 people is to be selected. How many different possible committees of 4 people can be selected from these 8 people if at least one of either Jane or Thomas is to be selected?

  28. Georgia license plates consist of three letters followed four numbers. How many license plate combinations are possible if the license plate contains no vowels (A, E, I, O, U, or Y) nor contains the number 5.

  29. A company issues a questionnaire whereby each employee must rank the 5 items with which he or she is most satisfied. The items are wages, work environment, vacation time, job security, supervisors, health insurance, break time, and retirement plan. The ranking is to be indicated by the numbers 1, 2, 3, 4 and 5, where 1 indicates the item involving the greatest satisfaction and 5 the least. In how many ways can an employee answer this questionnaire?

  30. A keypad lock has 10 different digits, and a sequence of 5 different digits must be selected for the lock to open. How many key pad combinations are possible?

  31. There are five women and six men in a group. From this group a committee of 4 is to be chosen. How many different ways can a committee be formed that contain three women and one man?

  32. How many 1 pair combinations are there in a standard deck of 52 cards?

  33. A class of 30 students enter a room with 40 desks. How many different ways can those students be seated?

  34. If a school has lockers with 50 numbers on each combination lock, how many possible combinations using three numbers are there?

  35. Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

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

  37. How many integers, greater than \(999\) but not greater than \(4000\), can be formed with the digits \(0\), \(1\), \(2\), \(3\) and \(4\) if repetition of digits is allowed?

  38. If a license plate has exactly \(3\) letters and \(4\) digits, how many license plates are possible if there must be exactly one vowel (excluding Y) and one \(7\) in the sequence?

  39. Proteins are made up of chains of amino acids. Insulin is a relatively small protein with \(53\) amino acid residues. How many possible proteins of length \(53\) can be made with \(20\) possible amino acids for each position in the protein?

  40. We wish to know how many color combinations can be made from four different colored marbles if we use only three marbles at a time. The marbles are colored red, green, white, and yellow.

  41. Two girls and their dates go to the drive-in, and each wants a different flavored ice cream cone. The drive-in has 24 flavors of ice cream. How many combinations of flavors may be chosen among the four of them if each one selects one flavor?

  42. We want to paint three rooms in a house, each a different color, and we may choose from seven different colors of paint. How many color combinations are possible for the three rooms?

  43. Find the number of ways to take 10 people and place them in 3 groups of 3 where order does not matter.

  44. Find the number of ways to take 20 objects and arrange them in groups of 5 at a time where order does not matter.

  45. Find the number of ways to arrange 5 objects that are chosen from a set of 7 different objects given that order matters.

  46. What is the total number of possible 5-letter arrangements of the letters WHITE if each letter is used exactly once?

  47. How many different 3-digit numerals can be made from the digits \(\{4,5,6,7,8\}\) if a digit can appear just once in a numeral?

  48. We wish to know how many color combinations can be made from four different colored marbles if we use only three marbles at a time. The marbles are colored red, green, white, and yellow.

  49. At the pizza place, there are 8 toppings that you can put on your pizza. If you can order any number of those 8 toppings, then how many different toppings could you possibly order?

5 Multinomial Counting Problems#

Formulas and Notation#

  • Multinomial Coefficients: \(\binom{n}{k_1,k_2,\dots,k_m}=\frac{n!}{k_1!k_2!\cdots k_m!}\) , given unordered groups of sizes \({k_1,k_2,\dots,k_n}\) such that \(\sum k_i=n\), the multinomial coefficient gives the number ways to arrange \(n\) objects into these groups.

  • If this is confusing, a string of binomial coefficients can be used instead, choosing each group one at a time rather than choosing all the groups at once.

Problems#

  1. Ten consultants will be divided into three teams of three and leader. How many different possibilities for the team configurations exist?

  2. A teacher divides a class of 20 students into five teams of 4. How many different possibilities exist?

  3. A teacher divides a class of 20 students into four teams of 5. How many different possibilities exist? Compare this to your answer to the previous question. Are there more or fewer possibilities than before?

  4. A dormitory hallway of 25 girls is selecting several groups to host socials. They need two groups of 4 girls, two groups of 3 girls, two groups of 2 girls and a group of 7 girls for the semester finale. How many possibilities exist?

  5. Compare the expressions. Calculate the total number of unique orderings of the letters DREADED. Then calculate the number of ways seven boys can be divided up into a team of 3, a team of 2 and a pair of boys to not be on either team.