C OUNTING M ETHOD

T HE A DDITION P RINCIPLE

I NCLUSION -E XCLUSION P RINCIPLE

T HE M ULTIPLICATION P RINCIPLE

E X 1) A DDING G ROUPS OF S TUDENTS 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: if there are 3 students in both classes

E X 2) 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. How many people subscribe to A but not to B? 2. How many subscribe to B but not to A ? 3. How many do not subscribe to either paper? 4. Draw a Venn diagram for the newspaper survey.

E X 3) The menu for a restaurant is listed as follows: How many different dinners consist of 1 appetizer, 1 main- course, and 1 beve- rage ?

E X 4) (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 ?

E X 5) Let X be an n -element set. (1) How many subsets of X are formed? (2) How many ordered pairs of ( A, B ) satisfy ?

P ERMUTATIONS 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

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 C OMBINATIONS

E X 6) F ROM 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.

E X 6) F ROM 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?

E X 7) COMBINATIONS If a fair coin is flipped 9 times, how many different ways are there to have 7 or more heads?

E X 8) P ERMUTATIONS (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?

E X 9)D ISTRIBUTION 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?

E X 10)D ISTRIBUTION 2 How many nonnegative integer solutions are there to the equation ? How many positive integer solutions are there to the equation ? How many solutions with ?

The number of distributions of r identical objects into n different places is or

E X 11)