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