1. Chapter 8 Counting Principles: Further Probability Topics Section 8.2 Combinations

2. Which Counting Technique? If the problem involves more than one category or repetition, use the multiplication principle of counting and multiply the number of choices for each category. If the problem involves more than one category or repetition, use the multiplication principle of counting and multiply the number of choices for each category. Within any one category, if the order of selection is important, use permutations. Within any one category, if the order of selection is important, use permutations. Within any one category, if the order of selection is not important, use combinations. Within any one category, if the order of selection is not important, use combinations.

3. A tennis squad has 5 members. The coach needs to select the first singles player and then a second singles player. In how many ways can he do this? Since the order in which the players are chosen is important, we will use a permutation to solve. 5 P 2 = 20

4. The same tennis coach needs to select a doubles team from his five players. How many different doubles team does he have to choose from? Let the five players be represented by A, B, C, D, E. Teams:ABACADAE Teams:ABACADAE BABCBDBE CACBCDCE DADBDCDE EAEBECED

5. Since the team with AB and BA is the same, as are others, we can eliminate all those teams that are repeats. In other words, since the order does not matter, we only write down a combination of the players one time. Since the team with AB and BA is the same, as are others, we can eliminate all those teams that are repeats. In other words, since the order does not matter, we only write down a combination of the players one time. Teams:ABACADAE BABCBDBE CACBCDCE DADBDCDE EAEBECED

6. Teams:ABACADAE BCBDBE CDCE DE We now have 10 different doubles teams. This is an example of a combination problem.

7. Combinations A subset of items listed without regard to order is called a combination. A subset of items listed without regard to order is called a combination. Like permutations, repetitions are not allowed in combinations. Like permutations, repetitions are not allowed in combinations. Clue words: group, committee, set, sample, team Clue words: group, committee, set, sample, team Combinations are denoted by the notation nCr or Combinations are denoted by the notation nCr or

9. In how many ways can you construct a 5-person committee out of 30 people? In how many ways can you construct a 5-person committee out of 30 people? You have five places left for eight stamps in your stamp book. How many different ways can you select the five to place in your stamp book? You have five places left for eight stamps in your stamp book. How many different ways can you select the five to place in your stamp book? There are 10 chips in a bag numbered from 1 to 10. Four chips are selected at random. How many different ways are there of selecting the four chips? There are 10 chips in a bag numbered from 1 to 10. Four chips are selected at random. How many different ways are there of selecting the four chips?

10. Suppose that three computer boards in a production run of forty are defective. A sample of four is to be selected and checked for defects. Suppose that three computer boards in a production run of forty are defective. A sample of four is to be selected and checked for defects. How many samples can be chosen? How many samples can be chosen? How many samples will contain at least one defective board? How many samples will contain at least one defective board? In how many ways can a sample of five chocolates be selected from a box of twenty- four chocolates? In how many ways can a sample of five chocolates be selected from a box of twenty- four chocolates?

11. Suppose you have a group of 10 children consisting of 4 girls and 6 boys. Suppose you have a group of 10 children consisting of 4 girls and 6 boys. How many four-person teams can be chosen that consist of two girls and two boys? How many four-person teams can be chosen that consist of two girls and two boys? How many four-person teams contain at least one girl? How many four-person teams contain at least one girl? In how many ways can a five-card hand consisting of three diamonds be dealt from a standard deck of 52 cards? In how many ways can a five-card hand consisting of three diamonds be dealt from a standard deck of 52 cards?

12. How many ways can a student choose eight questions from a twelve-question exam if at least three questions must be chosen from the first five and three questions from the last seven? How many ways can a student choose eight questions from a twelve-question exam if at least three questions must be chosen from the first five and three questions from the last seven? Two co-captains are to be selected from the starting five for a basketball team. In how many ways can this be done? Two co-captains are to be selected from the starting five for a basketball team. In how many ways can this be done? The student association each year selects a council consisting of 7 members. If there are 10 candidates for the 7-member council, how many different councils may be elected? The student association each year selects a council consisting of 7 members. If there are 10 candidates for the 7-member council, how many different councils may be elected? How many different poker hands can be dealt from a standard deck of 52 cards? How many different poker hands can be dealt from a standard deck of 52 cards?

13. How many committees can be selected from four teachers and 100 students if each committee must have two teachers and three students? How many committees can be selected from four teachers and 100 students if each committee must have two teachers and three students? If the Xerox Corporation has to transfer four of its 10 junior executives to a new location, in how many ways can the four executives be chosen? If the Xerox Corporation has to transfer four of its 10 junior executives to a new location, in how many ways can the four executives be chosen? A newspaper boy discovers while delivering his papers that he doesn’t have enough papers. He has eight houses left to deliver to, but only five papers left. In how many ways can he deliver the remaining newspapers? A newspaper boy discovers while delivering his papers that he doesn’t have enough papers. He has eight houses left to deliver to, but only five papers left. In how many ways can he deliver the remaining newspapers?

14. Alice has a penny, a nickel, a dime, a quarter, and a half- dollar. She may spend any three coins. Alice has a penny, a nickel, a dime, a quarter, and a half- dollar. She may spend any three coins. In how many ways can Alice do this? In how many ways can Alice do this? What is the most money she can spend using just three coins? What is the most money she can spend using just three coins? Joe has to take a math exam that consists of 10 questions. He must answer only seven of the 10 questions. Joe has to take a math exam that consists of 10 questions. He must answer only seven of the 10 questions. In how many ways can Joe choose the seven questions? In how many ways can Joe choose the seven questions? If he must answer the first and last questions and still answer a total of only seven, in how many ways can he do this? If he must answer the first and last questions and still answer a total of only seven, in how many ways can he do this?