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COUNTING METHOD THE ADDITION PRINCIPLE INCLUSION-EXCLUSION PRINCIPLE THE MULTIPLICATION PRINCIPLE EX1) ADDING GROUPS OF STUDENTS Dr. Abercrombie has 15 students in an abstract algebra class and 20 students in a linear algebra class. How many different students are in these two classes? Case1 : if there is no students in both classes Case2: classes if there are 3 students in both EX2) A city has two daily newspapers, A and B say. The following information was obtained from a survey of 100 city residents: 35 people subscribe to A, 60 subscribe to B, and 20 subscribe to both. 1. 2. 3. 4. How many people subscribe to A but not to B? How many subscribe to B but not to A? How many do not subscribe to either paper? Draw a Venn diagram for the newspaper survey. EX3) The menu for a restaurant is listed as follows: How many different dinners consist of 1 appetizer, 1 maincourse, and 1 beverage ? EX4) (1) How many strings of length 4 can be formed using the letters ABCDE if repetitions are not allowed? (2) How many strings of part(1) begin with letter B ? (3) How many strings of part(1) do not begin with letter B ? EX5) Let X be an n-element set. (1) How many subsets of X are formed? (2) How many ordered pairs of (A, B) satisfy A B X ? PERMUTATIONS A permutation of a set of objects is an arrangement of the objects in a specific order without repetition. P (n,r) — a permutation of n objects taken r at a time without repitition COMBINATIONS A combination of a set of n object taken r at a time without repetition is an r -element subset of the set of n objects. The arrangement of the elements in the subset does not matter. C (n,r ) — a combination of n objects taken r at a time without repetition EX6) FROM A COMMITTEE OF 10 PEOPLE CONSISTING OF 6 WOMEN AND 4 MEN, (1) In how many ways can we choose a chairperson, a vice-chairperson, and a secretary, assuming that one person cannot hold more than one position? (2) In how many ways can we choose a subcommittee of 3 people? In a permutation, order is vital. In a combination, order is irrelevant. EX6) FROM A COMMITTEE OF 10 PEOPLE CONSISTING OF 6 WOMEN AND 4 MEN, (3) In how many ways can we choose a subcommittee consisting of 3 women and 2 men? (4) In how many ways can we choose a subcommittee of equal number of women and men? (5) In how many ways can we choose a subcommittee of 5 people with at least 2 women? EX7)COMBINATIONS If a fair coin is flipped 9 times, how many different ways are there to have 7 or more heads? EX8) PERMUTATIONS (1) How many possibilities are there that everyone of 25 member of a group has a different birthday? (2) How many possibilities are there that at least two of 25 members of a group have a common birthday? EX9)DISTRIBUTION 1 How many routes are there from the lower-left corner of n × n square grid to the upper-right corner if we are restricted to traveling only to the right or upward? EX10)DISTRIBUTION 2 How many nonnegative integer solutions are there to the equation x y z? w 10 How many positive integer solutions are there to the equation x y z w ?10 How many solutions with x 1, y 2, z 3, w? 0 The number of distributions of r identical objects into n different places is C (n r 1, r ) or C (n r 1, n 1) EX11)