Permutations, Combinations and Probability



Permutations

Evaluate

1. 4P3 2. 7P5 3. 10P7 4. 10P3

5. How many 5 digit numbers can be named using the digits 5, 6, 7, 8, and 9 without repetition?, with repetition?

6. How many 4 digit numbers can be named using the digits 2, 3, 4, and 5 without repetition?, with repetition?

7. In how many ways can 5 students be arranged in a straight line?, in a circle?

8. In how many ways can 7 athletes be arranged in a straight line?, in a circle?

9. In how many distinguishable ways can the letters of the word DIGIT be arranged?

10. In how many ways distinguishable ways can the letters of the word RABBIT be arranged?

11. How many 7 digit phone numbers can be formed with the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, assuming that no digit is used more than once and the first digit is not 0?

12. A program is planned to have 5 rock numbers and 4 speeches. In how many ways can this be done if a rock number and a speech are to alternate and the rock numbers come first?

13. Suppose the expression a2b3c4 is rewritten without exponents. In how many ways can this be done?

14. Suppose the expression a3bc2 is rewritten without exponents. In how many ways can this be done?

15. In how many ways could King Arthur and his 12 knights sit at his round table?

16. In how many ways can 4 people be seated at a bridge table?

17. A penny, nickel, dime, quarter, and half-dollar are arranged in a straight line.

a. Considering just the coins, in how many ways can they be lined up?

b. Considering the coins and heads and tails, in how many ways can they be lined up?

18. A penny, nickel, dime and quarter are arranged in a straight line.

a. Considering just the coins, in how many ways can they be lined up?

b. Considering the coins and heads and tails, in how many ways can they be lined up?

19. Compute 52P4. 20. Compute 50P5.

21. A professor is going to grade her 24 students on a curve. She will give 3 A's, 5 B's, 9 C's, 4 D's, and 3 F's. In how many ways can she do this?

22. A professor is going to grade his 20 students on a curve. He will give 2 A's, 5 B's, 8 C's, 3 D's, and 2 F's. In how many ways can he do this?

23. How many distinguishable code symbols can be formed with the letters for the word MATH? BUSINESS? PHILOSOPHICAL?

24. How many distinguishable code symbols can be formed with the letters for the word ORANGE? BIOLOGY? MATHEMATICS?

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25. A state forms it license plates by first listing a number that corresponds to the county in which the car owner lives (the names of the counties are alphabetized and the number is its location in that order). Then the plate lists a letter of the alphabet, and this is followed by a number from 1 to 9999. How many such plates are possible if there are 80 counties?

26. How many code symbols can be formed using 4 out of 5 letters of A, B, C, D, and E if the letters:

a. are not repeated?

b. can be repeated?

c. are not repeated but must begin with D?

d. are not repeated but must end with DE?

Simplifying the following factorials:

27. 28.

29. 30.

31. 32.

Solve for x:

33. xP5 = 7xP4 34. xP4 = 8x – 1P3

35. xP5 = 9x – 1P4 36. xP4 = 8xP3

37. 14⋅xP3 = x + 2P4 38. xP5 = 18⋅x – 2P4

Answers

Permutations

1. 24 3. 604,800 5. 120; 3125

7. 120; 24 9. 60 11. 544,320

13. 1260 15. 479,001,600 17. a) 120 b) 3840

19. 6,497,400 21. 16,491,024,950,400 23. 24

25. 20,797,920 27. 29. n + 1

31. 33. 11 35 9

37. n = 5, 6

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