1. 1. In how many ways can six people sit in a six-passenger car?

2. 2. From a pool of twelve candidates, the offices of president, vice-president, secretary, and treasurer will be filled. In how many different ways can the offices be filled?

3. 3. In how many different orders can four girls and four boys walk through a doorway single file if the girls walk through before the boys?

4. 4. Three cards are drawn at random from an ordinary deck of 52 cards
4. Three cards are drawn at random from an ordinary deck of 52 cards. Find the number of ways in which a diamond is chosen, followed by a heart, followed by a diamond.

5. 5. How many numbers consisting of five different digits can be made from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 if the number must be odd?

6. 6. How many arrangements of the letters in the word TRIANGLE can be made that begin with T and end with E?

7. 7. Of seven Latin books and three English books, in how many ways can four Latin and one English book be placed on a shelf so that the English book is always in the middle?

8. 8. How many numbers less than 10,000 can be made with the eight digits 0, 1, 2, 3, 4, 5, 6, 7?

9. 9. How many five digit numbers begin with an odd digit and end with an even digit?

10. 10. A train, not counting the engine and caboose, consists of eight cars. If Car B must follow Car A, how many ways are there of putting the train together?

11. 11. In the old days, telephone numbers used to consist of two letters followed by five numbers. If “O” is not allowed for
the first letter and the two letters must be different, how many different telephone numbers could be arranged?

12. 12. A high school coach must decide on the batting order for a baseball team of nine players. How many different batting orders are possible if the pitcher always bats last, the catcher or first baseman bats eighth, and either the shortstop or left fielder bats first?

13. 13. How many automobile license plates can be made if each plate contains three different letters followed by three
different digits with the first digit being non-zero?

14. 14. In how many ways can seven persons sit in a row if three people insist on sitting together?

15. 15. Five boys and four girls are to be seated in a row containing nine seats. How many ways can this be done if they can sit anywhere?

16. 16. Five boys and four girls are to be seated in a row containing nine seats. How many ways can this be done if the sexes must alternate?

17. 17. Five boys and four girls are to be seated in a row containing nine seats. How many ways can this be done if the sexes must sit together?

18. 18. How many license plates are possible if the first two characters must be letters, at least one of which is a vowel, and the last four characters are digits, the first of which is non-zero?

19. 19. How many ways are there to arrange seven people for a photograph if three specific people must be in the middle?

20. 20. How many odd numbers of three digits can be formed without the repetition of any digit in a number, from the digits
1, 2, 3, 5, 8?

21. 21. How many numbers greater than 300 and less than 1000 can be made with the digits 1, 2, 3, 4, 5 if no digit is repeated in any number?

22. 22. How many numbers between 100 and 1000 can be written with the digits 0, 1, 2, 3, 4 if no digit is repeated in any number?

23. 23. In how many ways can three girls and three boys be seated in a row, if no two girls and no two boys are to occupy
adjacent seats?

24. 24. How many different radio stations can be named with either three different letters of the alphabet or four different
letters in which W must come first?

25. 25. How many four-digit numbers can be formed which are less than 5000?

26. 26. How many numbers less than 7000 contain no 3’s?

27. 27. In how many ways can six family members line up for a family portrait if Mom and Dad must be in the middle?

28. 28. How many positive integers between 999 and 5000 be formed using the digits 2, 3, 4, 5, 6, 7 if no digit may be repeated?

29. 29. How many arrangements of answers are possible on a multiple-choice test with five questions, each of which has four possible answers?

30. 30. How many numbers less than 100,000 can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 if the first digit cannot
be 0 or 1 and no digit may be repeated?