Counting Problems With Solutions
Counting problems are presented along with their detailed solutions and detailed explanations.
Counting PrincipleLet us start by introducing the counting principle using an example. A student has to take one course of physics, one of science and one of mathematics. He may choose one of 3 physics courses (P1, P2, P3), one of 2 science courses (S1, S2) and one of 2 mathematics courses (M1, M2). In how many ways can this student select the 3 courses he has to take?Let us use a tree diagram that shows all possible choices. The first column on the left shows the 3 possible choices of the physics course: P1, P2 or P3. Then the second column shows the 2 possible choices of the science course and the last column shows the 2 possible choices for the mathematics course. The different ways in which the 3 courses may be selected are: (P1 S1 M1), (P1 S1 M2), (P1 S2 M1), (P1 S2 M2) (P2 S1 M1), (P2 S1 M2), (P2 S2 M1), (P2 S2 M2) (P3 S1 M1), (P3 S1 M2), (P3 S2 M1), (P3 S2 M2) The total number of choices may be calculated as follows: Let n1 be the number of choices of the physics course, here n1 = 3. Let n2 be the number of choices of the science course, here n2 = 2. Let n3 be the number of choices of the mathematics course, here n3 = 2. It is clear from the tree diagram above that the total number N of choices may be calculated as follows: Using the above problem, we can generalize and write a formula related to counting as follows: "If events E1, E2, E3 ... can occur in n1, n2, n3 ... different ways respectively, the number of ways that all events can occur is equal to
Problem 1To buy a computer system, a customer can choose one of 4 monitors, one of 2 keyboards, one of 4 computers and one of 3 printers. Determine the number of possible systems that a customer can choose from.
Solution to Problem 1
Problem 2In a certain country telephone numbers have 9 digits. The first two digits are the area code (03) and are the same within a given area. The last 7 digits are the local number and cannot begin with 0. How many different telephone numbers are possible within a given area code in this country?
Solution to Problem 2
Problem 3A student can select one of 6 different mathematics books, one of 3 different chemistry books and one of 4 different science books. In how many different ways can a student select a book of mathematics, a book of chemistry and a book of science?
Solution to Problem 3
Problem 4There are 3 different roads from city A to city B and 2 different roads from city B to city C. In how many ways can someone go from city A to city C passing by city B?
Solution to Problem 4
Problem 5A man has 3 different suits, 4 different shirts and 5 different pairs of shoes. In how many different ways can this man wear a suit, a shirt and a pair of shoes?
Solution to Problem 5
Problem 6In a company , ID cards have 5 digit numbers.a) How many ID cards can be formed if repetition of the digit is allowed? b) How many ID cards can be formed if repetition of the digit is not allowed?
Solution to Problem 6
Problem 7In a certain country, licence plate numbers have 3 letters followed by 4 digits. How many different licence plate numbers can be formed? (letters and digits may be repeated).
Solution to Problem 7
Problem 8Using the digits 1, 2, 3 and 5, how many 4 digit numbers can be formed ifa) The first digit must be 1 and repetition of the digits is allowed? b) The first digit must be 1 and repetition of the digits is not allowed? b) The number must be divisible by 2 and repetition is allowed? b) The number must be divisible by 2 and repetition is not allowed?
Solution to Problem 8
Problem 9A coin is tossed three times. What is the total number of all possible outcomes?
Solution to Problem 9
Problem 10Two dice are rolled. What is the total number of all possible outcomes?
Solution to Problem 10
Problem 11A coin is tossed and a die is rolled. What is the total number of all possible outcomes?
Solution to Problem 11
More References and linkselementary statistics and probabilities. |