1. In how many different ways can the letters in the word STUDY be ordered?
2. Martha invited [latex]4[latex] friends to go with her to the movies. There are [latex]120[latex] different ways in which they can sit together in a row of [latex]5[latex] seats, one person per seat. In how many of those ways is Martha sitting in the middle seat?
3. How many [latex]3[latex]-digit positive integers are odd and do not contain the digit [latex]5[latex] ?
4. From a box of [latex]10[latex] lightbulbs, you are to remove [latex]4[latex]. How many different sets of [latex]4[latex] lightbulbs could you remove?
5. A talent contest has [latex]8[latex] 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 [latex]3[latex] prizes?
(b) How many different groups of [latex]3[latex] people can get prizes?
6. If an integer is randomly selected from all positive [latex]2[latex]-digit integers, what is the probability that the integer chosen has
(a) a [latex]4[latex] in the tens place?
(b) at least one[latex]4[latex] in the tens place or the units place?
(c) no [latex]4[latex] in either place?
7. There are [latex]5[latex] 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) [latex]1[latex]
(B) [latex]3[latex]
(C) [latex]6[latex]
(D) [latex]9[latex]
(E) [latex]12[latex]
8. Four pool balls—[latex]A, B, C, D[latex]—are randomly arranged in a straight line. What is the probability that the order will actually be [latex]A, B, C, D[latex] ?
(A) [latex]1/4[latex]
(B) [latex]\frac { 1 }{ { _{ 4 }{ C }_{ 4 } } }[latex]
(C) [latex]\frac { 1 }{ { _{ 4 }{ P }_{ 4 } } }[latex]
(D) [latex]1/2[latex]!
(E) [latex]1/3[latex]!
9. A basketball team has [latex]11[latex] players on its roster. Only [latex]5[latex] players can be on the court at one time. How many different groups of [latex]5[latex] players can the team put on the floor?
(A) [latex]{ 5 }^{ 11 }[latex]
(B) [latex]{ _{ 11 }{ C }_{ 5 } }[latex]
(C) [latex]{ _{ 11 }{ P }_{ 5 } }[latex]
(D) [latex]{ 11 }^{ 5 }[latex]!
(E) [latex]11[latex]!*[latex]5[latex]!
10. This is how Edward’s Lotteries work. First, [latex]9[latex] different numbers are selected. Tickets with exactly [latex]6[latex] of the [latex]9[latex] 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 [latex]6[latex] numbers drawn from the [latex]9[latex] randomly. There is exactly one winning ticket in the lottery system. How many tickets can the lottery system print?
(A) [latex]{ _{ 9 }{ P }_{ 6 } }[latex]
(B) [latex]{ _{ 9 }{ P }_{ 3 } }[latex]
(C) [latex]{ _{ 9 }{ C }_{ 9 } }[latex]
(D) [latex]{ _{ 9 }{ C }_{ 6 } }[latex]!
(E) [latex]{ 6 }^{ 9 }[latex]
11. How many different strings of letters can be made by reordering the letters of the word SUCCESS?
(A) [latex]20[latex]
(B) [latex]30[latex]
(C) [latex]40[latex]
(D) [latex]60[latex]
(E) [latex]420[latex]
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 [latex]4[latex]-digit positive integers where all [latex]4[latex] digits are less than [latex]5[latex] | [latex]625[latex] |
14. A state issues automobile license plates using two letters selected from a [latex]26[latex]-letter alphabet, as well as four numerals selected from the digits [latex]0[latex] through [latex]9[latex], inclusive. Repeats are permitted. For example, one license plate combination could be [latex]GF3352[latex].
| Quantity A | Quantity B |
| The number of possible unique license plate combination | [latex]6,000,000[latex] |
15. A small nation issues license plates that consist of just one number (selected from the digits [latex]0[latex] through [latex]9[latex], inclusive) and four letters, selected from a [latex]20[latex]-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) [latex]1,599,990[latex]
(B) [latex]1,599,999[latex]
(C) [latex]1,600,000[latex]
(D) [latex]4,569,759[latex]
(E) [latex]4,569,760[latex]
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
