Counting, Permutations and Combinations
1. In how many possible ways can a hockey team of 14 players elect a captain and two assistant captains from the group?
2. At a restaurant, a menu has a choice of 4 appetizers, 2 main courses and 3 desserts. Draw a tree diagram to determine how many different meals are available to a customer?
3. A club has a membership of 10 men and 12 women. In how many possible ways can the club select an executive of a president, vice president and 3 directors if:
a) the president must be a women and the vice-president a man?
b) the president and vice president cannot be both women or both men?
c) there must be at least two men on the executive?
4. How many different arrangements can be made from all the letters of the word distinct if:
a) there is no restriction?
b) the first and last letters must be the same?
6. How many different fruit baskets can be made from three bananas, six oranges, four apples, five kiwis, one pineapple, one cantaloupe and one mango?
7. In how many ways can three different awards be distributed among 20 students in the following situations:
a) no student can receive more than one award?
b) there is no limit on the number of awards won any student?
8. In how many ways can twenty clients be allocated evenly between:
a) ten employees?
b) five employees?
9. How many 5 letter arrangements of the letters of EQUATIONS are possible if there are:
a) no restrictions?
b) if QU must be together and Q must come before U?
10. In a school of 1052 students, how many ways can a dance committee of 5 students be chosen if
a) the committee includes Bob Smith and Brenda Jones?
b) The committee includes at least one of Bob Smith, Brenda Jones or Brian Chan?
11. The Swiss Embassy in Ottawa ha 100 employees: 39 speak German, 32 speak Italian, 47 speak French, 20 speak both German and French, 10 speak both Italian and German, 15 speak both Italian and French and 5 speak all three languages. Draw a Venn Diagram to determine how many people do not speak any of these languages.
12. A package of 20 transistors contains fifteen that are perfect and five that are defective. In how many ways can five of the transistors be selected so that at least three are perfect?
13. A hockey team has 19 players. If 2 defense-players can be selected from the players that play defense in 28 ways, how many players do not play in the defense position? A complete algebraic solution must be shown.
14. Simplify the following by first expressing them in factorial notation. DO NOT Evaluate
a) P(10, 7) b) C(5, 3) c)
18. Simplify the following:
a) b)
19. In how many ways can a 5-card poker hand be selected if exactly two aces must be selected?
20. How many 5-card poker hands can be dealt each having a single ace and a single king?
21. In how many ways can a 13-card bridge hand be dealt if it has 4 spades, 3 hearts, 2 clubs and 4 diamonds?
22. In how many ways can a 13-card bridge hand be dealt if it has the Ace and King of Spades, 3 other spades, 4 hearts, 3 clubs and the Queen of Diamonds?
23. How many three-letter “words” can be formed using the letters taken from the word SILLY?