1 2.3 Counting Sample Points (6)(6)(6)= 216 Example: How many outcome sequences are possible when a die is rolled three times?

2 Multiplication Rule (Theorem 2.1) If an operation can be performed n 1 ways, and if for each of these a second operation can be performed n 2 ways, then two operations can be performed together in n 1 n 2 ways.

3 Theorem 2.2 If an operation can be performed in n 1 ways, and if for each of these a second operation can be performed n 2 ways, and for each of the first two a third operation can be performed in n 3 ways, and so forth, then the sequence of k operations can be performed together in n 1 n 2 … n k ways. Example: A computer system uses passwords that consist of five letters followed by a single digit. How many passwords are possible? 26 5 (10) Read Example 2.15 and Example 2.16, page 32-33

4 Permutation: A permutation is an arrangement of a set of objects in a definite order. Example: The letters a, b, and c have 6 permutations. Theorem 2.3 The number of permutations of n distinct objects is n! (called n factorial) Theorem 2.4 The number of permutations of n distinct objects taken r at a time is = n (n-1)…(n - r +1)

5 Example 2.17: Three different awards will be given for a class of 25 graduate students in a math department. If each student can receive at most one award, how many possible ways are there? = 13, 800

6 Theorem 2.6 Suppose a set of n objects contains n 1 individuals of type 1, n 2 individuals of type 2, ……, n k individuals of type k. Then the number of distinct permutations is

7 Example (Investing in Stock) :Consider the letters BBBBBAAAA which represent a sequence of recent years in which the Dow Jones Industrial Average was below (B) the mean or above (A) the mean. How many ways can we arrange the letters BBBBBAAAA? Is there a pattern suggesting that it would be wise to invest in stocks? BBBBBAAAA: All below values occur at the beginning, and all above values occur at the end, it suggest an increasing in stock values. 9!/5!4! = 126

8 Example: An item in a store is labeled by printing with four thick lines, three medium lines, and two thin lines. If each ordering of the nine lines represents a different label, how many different labels can be generated by using the scheme? 9!/4!3!2! = 1260.

9 Combination: A combination is a selection of object without regard of order. Theorem 2.8 The number of combinations of n distinct objects taken r at a time is

10 Example 2.21: A firm offers a choice of 10 free software packages to buyers of their new home computers. There are 25 packages from which to choose. (a) In how many ways can the selection be made? (b) If five of the packages are computer games. How many selections are possible if exactly three computer games are selected?

11 Theorem A set of n objects contains n 1 individuals of type 1, n 2 individuals of type 2, …, n k individuals of type k. If we randomly select a set of r objects which contains r 1 type 1, r 2 type 2, …, r k type k, then the total number of possible ways of selecting the set is complement