Section Exercises

For the following exercises, assume that there are [latex]n[/latex] ways an event [latex]A[/latex] can happen, [latex]m[/latex] ways an event [latex]B[/latex] can happen, and that [latex]A\text{ and }B[/latex] are non-overlapping. 1. Use the Addition Principle of counting to explain how many ways event [latex]A\text{ or }B[/latex] can occur. 2. Use the Multiplication Principle of counting to explain how many ways event [latex]A\text{ and }B[/latex] can occur. Answer the following questions. 3. When given two separate events, how do we know whether to apply the Addition Principle or the Multiplication Principle when calculating possible outcomes? What conjunctions may help to determine which operations to use? 4. Describe how the permutation of [latex]n[/latex] objects differs from the permutation of choosing [latex]r[/latex] objects from a set of [latex]n[/latex] objects. Include how each is calculated. 5. What is the term for the arrangement that selects [latex]r[/latex] objects from a set of [latex]n[/latex] objects when the order of the [latex]r[/latex] objects is not important? What is the formula for calculating the number of possible outcomes for this type of arrangement? For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. 6. Let the set [latex]A=\left\{-5,-3,-1,2,3,4,5,6\right\}[/latex]. How many ways are there to choose a negative or an even number from [latex]\mathrm{A?}[/latex] 7. Let the set [latex]B=\left\{-23,-16,-7,-2,20,36,48,72\right\}[/latex]. How many ways are there to choose a positive or an odd number from [latex]A?[/latex] 8. How many ways are there to pick a red ace or a club from a standard card playing deck? 9. How many ways are there to pick a paint color from 5 shades of green, 4 shades of blue, or 7 shades of yellow? 10. How many outcomes are possible from tossing a pair of coins? 11. How many outcomes are possible from tossing a coin and rolling a 6-sided die? 12. How many two-letter strings—the first letter from [latex]A[/latex] and the second letter from [latex]B-[/latex] can be formed from the sets [latex]A=\left\{b,c,d\right\}[/latex] and [latex]B=\left\{a,e,i,o,u\right\}?[/latex] 13. How many ways are there to construct a string of 3 digits if numbers can be repeated? 14. How many ways are there to construct a string of 3 digits if numbers cannot be repeated? For the following exercises, compute the value of the expression. 15. [latex]P\left(5,2\right)[/latex] 16. [latex]P\left(8,4\right)[/latex] 17. [latex]P\left(3,3\right)[/latex] 18. [latex]P\left(9,6\right)[/latex] 19. [latex]P\left(11,5\right)[/latex] 20. [latex]C\left(8,5\right)[/latex] 21. [latex]C\left(12,4\right)[/latex] 22. [latex]C\left(26,3\right)[/latex] 23. [latex]C\left(7,6\right)[/latex] 24. [latex]C\left(10,3\right)[/latex] For the following exercises, find the number of subsets in each given set. 25. [latex]\left\{1,2,3,4,5,6,7,8,9,10\right\}[/latex] 26. [latex]\left\{a,b,c,\dots ,z\right\}[/latex] 27. A set containing 5 distinct numbers, 4 distinct letters, and 3 distinct symbols 28. The set of even numbers from 2 to 28 29. The set of two-digit numbers between 1 and 100 containing the digit 0 For the following exercises, find the distinct number of arrangements. 30. The letters in the word "juggernaut" 31. The letters in the word "academia" 32. The letters in the word "academia" that begin and end in "a" 33. The symbols in the string #,#,#,@,@,$,$,$,%,%,%,% 34. The symbols in the string #,#,#,@,@,$,$,$,%,%,%,% that begin and end with "%" 35. The set, [latex]S[/latex] consists of [latex]\text{900,000,000}[/latex] whole numbers, each being the same number of digits long. How many digits long is a number from [latex]S?[/latex] (Hint: use the fact that a whole number cannot start with the digit 0.) 36. The number of 5-element subsets from a set containing [latex]n[/latex] elements is equal to the number of 6-element subsets from the same set. What is the value of [latex]n?[/latex] (Hint: the order in which the elements for the subsets are chosen is not important.) 37. Can [latex]C\left(n,r\right)[/latex] ever equal [latex]P\left(n,r\right)?[/latex] Explain. 38. Suppose a set [latex]A[/latex] has 2,048 subsets. How many distinct objects are contained in [latex]A?[/latex] 39. How many arrangements can be made from the letters of the word "mountains" if all the vowels must form a string? 40. A family consisting of 2 parents and 3 children is to pose for a picture with 2 family members in the front and 3 in the back.

1. How many arrangements are possible with no restrictions?
2. How many arrangements are possible if the parents must sit in the front?
3. How many arrangements are possible if the parents must be next to each other?

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