1. Section 8.3 - Combinations
(with and without restrictions)

2. 1. In how many ways can a committee of 5 be chosen from a group of 10?

3. 2. From a group of 7 people, how many different committees can be formed if there must be at least three people on
the committee?

4. 3. Joel must answer 7 out of 10 questions on the English exam
3. Joel must answer 7 out of 10 questions on the English exam. If he must answer at least three out of the first four questions, how many different choices can he make for the exam?

5. 4. From a group of 20 sophomores and 10 freshmen, how many committees of four sophomores and three freshmen are possible if one of three specific sophomores must be on the committee?

6. 5. For next year’s starting basketball lineup, Mr
5. For next year’s starting basketball lineup, Mr. Sheets can choose either of two players for center, from five possibilities for two forwards, and from six possibilities for two guards. However, forwards Peter and Sean cannot be in the lineup at the same time. How many starting lineups are possible?

7. 6. A bridge hand consists of 13 cards
6. A bridge hand consists of 13 cards. How many bridge hands exist in which there are at least 3 aces and at least 3 kings? Leave your answer in terms of combinations.

8. 7. From nine different books including just two books of poetry, how many groups of five books each can be formed, if each group is to involve just one book of poetry?

9. 8. In how many ways can debating teams of three men each be chosen from seven candidates?

10. 9. A bridge hand consists of 13 cards dealt from a deck of 52
9. A bridge hand consists of 13 cards dealt from a deck of 52. In how many ways can a person be dealt a bridge hand
in which there are exactly 4 aces, 3 kings, 2 queens, and 2 jacks?

11. 10. In how many ways can a committee of three be chosen from eight persons so that
a. John is always included?
b. John is always excluded?

12. 11. Five cards consisting of an ace, a king, a queen, a jack, and a ten can be divided into two piles, one containing two
cards and the other three cards in how many ways?

13. 12. Six cards consisting of an ace, a king, a queen, a jack, a 10, and a 9 are to be divided into two groups of three each. In how many ways can the groups be formed?

14. 13. From seven sophomores and four juniors a committee of six is to be formed. In how many ways can this be done
when the committee consists of at least two juniors?

15. 14. In how many ways can a hand consisting of five spades, six hearts, and two other cards be selected? Leave your
answer in combination form.

16. 15. In how many ways is it possible to draw a sum of money from a bag containing a dollar, a half dollar, a quarter, a dime, a nickel, and a penny?

17. 16. In how many ways can six students form themselves into three teams of two?

18. 17. In how many ways can 12 students be divided into three groups of two and two groups of three if Ken must be in a twosome?

19. 18. Students are required to answer 10 out of 15 questions on an exam
18. Students are required to answer 10 out of 15 questions on an exam. Of the 10 they answer, they must answer at least two of the first three problems. How many ways can a student answer 10 questions on the exam?

20. 19. How many 13 card hands having exactly 11 cards from any suit can be dealt?

21. 20. How many ways can a committee be formed of at least five students from a group of eight students?

22. 21. On the TV game show Wheel of Fortune, contestants are given the letters R, S, T, L, N, and E in the bonus round to help solve a puzzle. Contestants then get to choose three consonants and one vowel for addition help in solving the puzzle. In how many ways can a contestant do this?

23. 22. You are dealt five cards from an ordinary deck of 52 playing cards
22. You are dealt five cards from an ordinary deck of 52 playing cards. In how many ways can you get a full house (three of one kind and two of another kind.)

24. 23. The school is going to buy vehicles from a group of seven vans and four busses. In how many ways can this be done if the school buys five vehicles of which at least three are busses?

25. 24. Seven cards are chosen from a deck of 52 cards
24. Seven cards are chosen from a deck of 52 cards. Note: Leave your answer in combination form.
In how many ways can the seven cards include no red cards?

26. 25. Seven cards are chosen from a deck of 52 cards
25. Seven cards are chosen from a deck of 52 cards. Note: Leave your answer in combination form.
In how many ways can the seven cards include at least five black cards?

27. 26. Seven cards are chosen from a deck of 52 cards
26. Seven cards are chosen from a deck of 52 cards. Note: Leave your answer in combination form.
In how many ways can the seven cards include at least one red card?

28. 27. Seven cards are chosen from a deck of 52 cards
27. Seven cards are chosen from a deck of 52 cards. Note: Leave your answer in combination form.
In how many ways can the seven cards include four jacks and four kings?

29. 28. Seven cards are chosen from a deck of 52 cards
28. Seven cards are chosen from a deck of 52 cards. Note: Leave your answer in combination form.
In how many ways can the seven cards include four aces?

30. 29. Seven cards are chosen from a deck of 52 cards
29. Seven cards are chosen from a deck of 52 cards. Note: Leave your answer in combination form.
In how many ways can the seven cards include at least one queen?

31. 30. An urn contains seven white balls and three red balls
30. An urn contains seven white balls and three red balls. Three balls are selected. In how many ways can the three balls be drawn from the total of 10 balls:
if two balls are white and one is red?

32. 31. An urn contains seven white balls and three red balls
31. An urn contains seven white balls and three red balls. Three balls are selected. In how many ways can the three balls be drawn from the total of 10 balls:
if all three balls are white?

33. 32. An urn contains seven white balls and three red balls
32. An urn contains seven white balls and three red balls. Three balls are selected. In how many ways can the three balls be drawn from the total of 10 balls:
if all three balls are red?

34. 33. Consider a 12-card deck consisting of Jacks, Queens, and Kings of all suits.
Find the number of three-card hands possible.

35. 34. Consider a 12-card deck consisting of Jacks, Queens, and Kings of all suits.
Find the number of three-card hands that consist of 3 of a kind (i.e., all Jacks).

36. 35. Consider a 12-card deck consisting of Jacks, Queens, and Kings of all suits.
Find the number of three-card hands with a flush which are possible (i.e. all of the same suit).

37. 36. Consider a 12-card deck consisting of Jacks, Queens, and Kings of all suits.
Find the number of three-card hands which contain a pair (but not 3 of a kind).

38. 37. Consider a 12-card deck consisting of Jacks, Queens, and Kings of all suits.
Find the number of three-card hands in which there is no pair.

39. 38. A company employs 10 people.
If three are to be assigned to the executive suite, three to the marketing department, and four to a general pool of employees, in how many different ways can they be assigned?

40. 39. A company employs 10 people.
If the three in the executive suite are to be assigned to the president, executive vice-president, and financial vice-president, with the remaining seven assigned as in problem 38, in how many different ways can they now be assigned?

41. 40. A company employs 10 people.
If the three in the executive suite are to be assigned as in problem 39, the three in the marketing department are to be assigned to the general manager, manager for domestic marketing and manager for foreign marketing, and the remain four to a general pool of employees, in how many different ways can they be assigned?

42. 41. In a small college the number of professors in the life, social, and management sciences is four, five and seven respectively. A committee of six is to be chosen from this group of sixteen individuals.
How many possible committees are there?

43. 42. In a small college the number of professors in the life, social, and management sciences is four, five and seven respectively. A committee of six is to be chosen from this group of sixteen individuals.
If the committee is to consist of one, two, and three people, from the life, social, and management sciences departments, respectively, in how many ways can the committee be formed?

44. 43. In a small college the number of professors in the life, social, and management sciences is four, five and seven respectively. A committee of six is to be chosen from this group of sixteen individuals.
If the only restriction is that there be three people from the management sciences department, how many different committees can be formed?

45. 44. A shipment of 80 television sets is known to contain 10 that are defective according to some specifications. Suppose that 15 sets are selected at random from the 80. Note: Leave your answer in combination form.
How many different selections are possible?

46. 45. A shipment of 80 television sets is known to contain 10 that are defective according to some specifications. Suppose that 15 sets are selected at random from the 80. Note: Leave your answer in combination form.
In how many ways is it possible for none of the sets selected to be defective?

47. 46. A shipment of 80 television sets is known to contain 10 that are defective according to some specifications. Suppose that 15 sets are selected at random from the 80. Note: Leave your answer in combination form.
In how many ways is it possible for the sample of 15 sets to contain all the defective ones?

48. 47. A shipment of 80 television sets is known to contain 10 that are defective according to some specifications. Suppose that 15 sets are selected at random from the 80. Note: Leave your answer in combination form.
How many different samples of 15 sets are possible which contain exactly three defective ones?

49. 48. Four people are to be selected at random from a group of four couples. In how many ways can this be done under the following conditions:
There are no restrictions.

50. 49. Four people are to be selected at random from a group of four couples. In how many ways can this be done under the following conditions:
The group must have at least one couple.

51. 50. Four people are to be selected at random from a group of four couples. In how many ways can this be done under the following conditions:
Each couple must be represented in the group