1. In how many different ways can the letters in the word STUDY be ordered?
2. Martha invited \(\)4\(\) friends to go with her to the movies. There are \(\)120\(\) different ways in which they can sit together in a row of \(\)5\(\) seats, one person per seat. In how many of those ways is Martha sitting in the middle seat?
3. How many \(\)3\(\)-digit positive integers are odd and do not contain the digit \(\)5\(\) ?
4. From a box of \(\)10\(\) lightbulbs, you are to remove \(\)4\(\). How many different sets of \(\)4\(\) lightbulbs could you remove?
5. A talent contest has \(\)8\(\) contestants. Judges must award prizes for first, second, and third places, with no ties.
(a) In how many different ways can the judges award the \(\)3\(\) prizes?
(b) How many different groups of \(\)3\(\) people can get prizes?
6. If an integer is randomly selected from all positive \(\)2\(\)-digit integers, what is the probability that the integer chosen has
(a) a \(\)4\(\) in the tens place?
(b) at least one\(\)4\(\) in the tens place or the units place?
(c) no \(\)4\(\) in either place?
7. There are \(\)5\(\) doors to a lecture room. Two are red and the others are green. In how many ways can a lecturer enter the room and leave the room from different colored doors?
(A) \(\)1\(\)
(B) \(\)3\(\)
(C) \(\)6\(\)
(D) \(\)9\(\)
(E) \(\)12\(\)
8. Four pool balls—\(\)A, B, C, D\(\)—are randomly arranged in a straight line. What is the probability that the order will actually be \(\)A, B, C, D\(\) ?
(A) \(\)1/4\(\)
(B) \(\)\frac { 1 }{ { _{ 4 }{ C }_{ 4 } } }\(\)
(C) \(\)\frac { 1 }{ { _{ 4 }{ P }_{ 4 } } }\(\)
(D) \(\)1/2\(\)!
(E) \(\)1/3\(\)!
9. A basketball team has \(\)11\(\) players on its roster. Only \(\)5\(\) players can be on the court at one time. How many different groups of \(\)5\(\) players can the team put on the floor?
(A) \(\){ 5 }^{ 11 }\(\)
(B) \(\){ _{ 11 }{ C }_{ 5 } }\(\)
(C) \(\){ _{ 11 }{ P }_{ 5 } }\(\)
(D) \(\){ 11 }^{ 5 }\(\)!
(E) \(\)11\(\)!*\(\)5\(\)!
10. This is how Edward’s Lotteries work. First, \(\)9\(\) different numbers are selected. Tickets with exactly \(\)6\(\) of the \(\)9\(\) numbers randomly selected are printed such that no two tickets have the same set of numbers. Finally, the winning ticket is the one containing the \(\)6\(\) numbers drawn from the \(\)9\(\) randomly. There is exactly one winning ticket in the lottery system. How many tickets can the lottery system print?
(A) \(\){ _{ 9 }{ P }_{ 6 } }\(\)
(B) \(\){ _{ 9 }{ P }_{ 3 } }\(\)
(C) \(\){ _{ 9 }{ C }_{ 9 } }\(\)
(D) \(\){ _{ 9 }{ C }_{ 6 } }\(\)!
(E) \(\){ 6 }^{ 9 }\(\)
11. How many different strings of letters can be made by reordering the letters of the word SUCCESS?
(A) \(\)20\(\)
(B) \(\)30\(\)
(C) \(\)40\(\)
(D) \(\)60\(\)
(E) \(\)420\(\)
12. A 10-student class is to choose a president, vice president, and secretary from the group. Assuming that no person can occupy more than one post, in how many ways can this be accomplished?
13.
| Quantity A | Quantity B |
| The number of \(\)4\(\)-digit positive integers where all \(\)4\(\) digits are less than \(\)5\(\) | \(\)625\(\) |
14. A state issues automobile license plates using two letters selected from a \(\)26\(\)-letter alphabet, as well as four numerals selected from the digits \(\)0\(\) through \(\)9\(\), inclusive. Repeats are permitted. For example, one license plate combination could be \(\)GF3352\(\).
| Quantity A | Quantity B |
| The number of possible unique license plate combination | \(\)6,000,000\(\) |
15. A small nation issues license plates that consist of just one number (selected from the digits \(\)0\(\) through \(\)9\(\), inclusive) and four letters, selected from a \(\)20\(\)-letter alphabet. Repeats are permitted. However, there is one four-letter combination that is not allowed to appear on license plates. How many allowable license plate combinations exist?
(A) \(\)1,599,990\(\)
(B) \(\)1,599,999\(\)
(C) \(\)1,600,000\(\)
(D) \(\)4,569,759\(\)
(E) \(\)4,569,760\(\)
Answer:
1 5!=120
2 24
3 288
4 210
5 (a) 336 (b) 56
6 (a) 1/9 (b) 1/5 (c)4/5
7 E
8 C
9 B
10 D
11 E
12 720
13 B
14 A
15 A
