1. Topics to be covered: Produce all combinations and permutations of sets. Calculate the number of combinations and permutations of sets of m items taken n at a time. Apple basic fundamental counting principles such as The Pigeonhole Principle, Multiplication Principle, Addition Principle, and Binomial Theorem to practical problems. Solve probability problems such as conditional probability, probability of simple events, mutually exclusive events, and independent events. Find the odds that an event will occur given the probabilities and vice versa.

2.  A tree can be used to keep track of all the possibilities in a situation.  Example: A computer installation has 4 input/output units (A, B, C and D) and 3 CPU’s (X, Y, and Z). How many ways are there to pair an I/O unit with a CPU? The total number of ways to pair the two types is the same as the number of branches in the tree, which is: 3 + 3 + 3 + 3 = 4 * 3 = 12 A B C D X Y Z X Y Z X Y Z X Y Z

3.  If an operation consists of k steps and The first step can be performed in n 1 ways, The second can be performed in n 2 ways (regardless of how the first step was performed), …. The kth step can be performed in n k ways (regardless of how the preceding steps were performed),  Then the entire operation can be performed in n 1 n 2 …n k ways.

4.  A permutation of a set of objects is an ordering of the objects in a row.  Example: The set of alements {a, b, c} has six permutations: abc acb cba bac bca cab  There are n ways to perform step one, (n – 1) ways to perform step two, so the number of permutations can be given by… n(n – 1)(n – 2)….2 * 1 = n!

5.  Given the set {a, b, c} there are six ways to select two letters from the set and write them in order. ab ac ba bc ca cb  An r-permutation of a set of n elements is an ordered selection of r elements taken from the set of n elements.  We use the notation n P r  Answer given by the formula: n P r =

6.  How many different ways can three of the letters of the words BYTES can be chosen and written in a row?  There are 5 possible letters, and you are selecting 3 of them, so this is written as 5 P 3  5 P 3 =  So, there are 60 possible permutations.

7.  The addition rule is used to calculate the number of elements in the union, difference, or intersection of mutually disjoint finite sets.  Suppose a finite set A equals the union of k mutually disjoint subsets A 1, A 2,..., A k. Then…  N(A) = N(A 1 ) + N(A 2 ) +... + N(A k )

8.  A password consists of 1-3 letters chosen from the 26 in the alphabet with repetitions allowed.  The set of all passwords can be partitioned into subsets consisting of those of length 1, length 2, and length 3. Passwords of length 1 = 26 Passwords of length 1 = 26 2 Passwords of length 1 = 26 3  So by the addition rule the total number of passwords = 26 + 26 2 + 26 3 = 18,278

9.  An r-combination of a set of n elements is a subset of r of the n elements.  Notation: n C r  n C r =

10.  How many distinct 5 person teams can be chosen from a group of 12? Solution:  What if two members refuse to work together? (C and D) Teams that contain C but not D: 10 C 4 = 210 Teams that contain D but not C: 10 C 4 = 210 Teams that contain neither C nor D: 10 C 5 = 252 Use the addition rule to solve…. 210 + 210 + 252 = 672

11.  The binomial theorem uses combinations to expand binomial powers:  Example: expand (a + b) 5

12.  The pigeonhole principle states that if n pigeons fly into m pigeonholes and n > m then at least one hole must contain two or more pigeons.  Example: In a group of thirteen people, must there be two who were born in the same month? Yes. Think of the 13 people as pigeons, and the 12 months as the pigeonholes. If you draw an arrow from every person to a month, then there must be at least one month that has two arrows pointing to it.

13.  A sample space is the set of all possible outcomes of a random process.  An event is a subset of a sample space.  A simple event is an event that consists of one outcome.  If S is a finite sample space in which all outcomes are equally likely and E is an event in S then the probability of E, denoted P(E), is  Note: for any finite set, N(A) denotes the number of elements in A.  The formula then becomes:

14.  The sample space for these questions is a standard 52 card deck. What is the probability that a black face card will be drawn on the first draw?  Answer: What is the probability that you will draw a 5 and then a face card?  Answer:

15.  Suppose that a couple has two children, both of which are equally likely to be a boy or a girl. 4 possible combinations: BB BG GB GG  Now suppose you are given the information that one is a boy. What is the probability that the other is a boy? Now there are only 3 possible combinations: BB BG GB Answer:  Formula for conditional probability:  The left side is read as “The probability of event B given event A.”

16.  Two event are independent if  Equivalently  Example: A coin is tossed and a single 6-sided die is rolled. Find the probability of the coin being heads and rolling a 3. P(heads) = 1/2 P(rolling 3) = 1/6 P(heads and 3) = 1/2 * 1/6 = 1/12

17.  Two events are considered mutually exclusive if they cannot occur at the same time (have no common outcomes)  Example: tossing a coin. It can result in heads or tails, but not both.