{"id":2331,"date":"2025-08-13T00:10:34","date_gmt":"2025-08-13T00:10:34","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2331"},"modified":"2026-04-01T08:04:59","modified_gmt":"2026-04-01T08:04:59","slug":"probability-and-counting-theory-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/probability-and-counting-theory-get-stronger\/","title":{"raw":"Probability and Counting Theory: Get Stronger","rendered":"Probability and Counting Theory: Get Stronger"},"content":{"raw":"

Counting Principles<\/h1>\r\n3. 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?\r\n\r\nFor the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.\r\n\r\n7. 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]\r\n\r\n9. How many ways are there to pick a paint color from 5 shades of green, 4 shades of blue, or 7 shades of yellow?\r\n\r\n11. How many outcomes are possible from tossing a coin and rolling a 6-sided die?\r\n\r\n13. How many ways are there to construct a string of 3 digits if numbers can be repeated?\r\n\r\nFor the following exercises, compute the value of the expression.\r\n\r\n15. [latex]P\\left(5,2\\right)[\/latex]\r\n\r\n17. [latex]P\\left(3,3\\right)[\/latex]\r\n\r\n21. [latex]C\\left(12,4\\right)[\/latex]\r\n\r\n23. [latex]C\\left(7,6\\right)[\/latex]\r\n\r\nFor the following exercises, find the distinct number of arrangements.\r\n\r\n31. The letters in the word \"academia\"\r\n\r\n33. The symbols in the string #,#,#,@,@,$,$,$,%,%,%,%\r\n\r\n35.\u00a0The 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:<\/em> use the fact that a whole number cannot start with the digit 0.)\r\n\r\n37. Can [latex]C\\left(n,r\\right)[\/latex] ever equal [latex]P\\left(n,r\\right)?[\/latex] Explain.\r\n\r\n41. A cell phone company offers 6 different voice packages and 8 different data packages. Of those, 3 packages include both voice and data. How many ways are there to choose either voice or data, but not both?\r\n\r\n43. A wholesale T-shirt company offers sizes small, medium, large, and extra-large in organic or non-organic cotton and colors white, black, gray, blue, and red. How many different T-shirts are there to choose from?\r\n\r\n45. An art store has 4 brands of paint pens in 12 different colors and 3 types of ink. How many paint pens are there to choose from?\r\n\r\n47. How many ways can a baseball coach arrange the order of 9 batters if there are 15 players on the team?\r\n\r\n49. A motorcycle shop has 10 choppers, 6 bobbers, and 5 caf\u00e9 racers\u2014different types of vintage motorcycles. How many ways can the shop choose 3 choppers, 5 bobbers, and 2 caf\u00e9 racers for a weekend showcase?\r\n\r\n51. Just-For-Kicks Sneaker Company offers an online customizing service. How many ways are there to design a custom pair of Just-For-Kicks sneakers if a customer can choose from a basic shoe up to 11 customizable options?\r\n\r\n53. Susan bought 20 plants to arrange along the border of her garden. How many distinct arrangements can she make if the plants are comprised of 6 tulips, 6 roses, and 8 daisies?\r\n

Binomial Theorem<\/h1>\r\n1. What is a binomial coefficient, and how it is calculated?\r\n\r\nFor the following exercises, evaluate the binomial coefficient.\r\n\r\n5. [latex]\\left(\\begin{array}{c}6\\\\ 2\\end{array}\\right)[\/latex]\r\n\r\n9. [latex]\\left(\\begin{array}{c}10\\\\ 9\\end{array}\\right)[\/latex]\r\n\r\nFor the following exercises, use the Binomial Theorem to expand each binomial.\r\n\r\n13. [latex]{\\left(4a-b\\right)}^{3}[\/latex]\r\n\r\n15. [latex]{\\left(3a+2b\\right)}^{3}[\/latex]\r\n\r\n19. [latex]{\\left(4x - 3y\\right)}^{5}[\/latex]\r\n\r\n21. [latex]{\\left({x}^{-1}+2{y}^{-1}\\right)}^{4}[\/latex]\r\n\r\nFor the following exercises, use the Binomial Theorem to write the first three terms of each binomial.\r\n\r\n23. [latex]{\\left(a+b\\right)}^{17}[\/latex]\r\n\r\n25. [latex]{\\left(a - 2b\\right)}^{15}[\/latex]\r\n\r\n27. [latex]{\\left(3a+b\\right)}^{20}[\/latex]\r\n\r\n29. [latex]{\\left({x}^{3}-\\sqrt{y}\\right)}^{8}[\/latex]\r\n\r\nFor the following exercises, find the indicated term of each binomial without fully expanding the binomial.\r\n\r\n33. The eighth term of [latex]{\\left(7+5y\\right)}^{14}[\/latex]\r\n\r\n35. The fifth term of [latex]{\\left(x-y\\right)}^{7}[\/latex]\r\n\r\n39. The eighth term of [latex]{\\left(\\frac{y}{2}+\\frac{2}{x}\\right)}^{9}[\/latex]\r\n\r\n45. In the expansion of [latex]{\\left(5x+3y\\right)}^{n}[\/latex], each term has the form [latex]\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right){a}^{n-k}{b}^{k}, \\text{where } k[\/latex] successively takes on the value [latex]0,1,2,...,n[\/latex]. If [latex]\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right)=\\left(\\begin{array}{c}7\\\\ 2\\end{array}\\right)[\/latex], what is the corresponding term?\r\n\r\n49.\u00a0Which expression cannot be expanded using the Binomial Theorem? Explain.\r\n
    \r\n \t
  • [latex]\\left({x}^{2}-2x+1\\right)[\/latex]<\/li>\r\n \t
  • [latex]{\\left(\\sqrt{a}+4\\sqrt{a}-5\\right)}^{8}[\/latex]<\/li>\r\n \t
  • [latex]{\\left({x}^{3}+2{y}^{2}-z\\right)}^{5}[\/latex]<\/li>\r\n \t
  • [latex]{\\left(3{x}^{2}-\\sqrt{2{y}^{3}}\\right)}^{12}[\/latex]<\/li>\r\n<\/ul>\r\n

    Probability<\/h1>\r\nFor the following exercises, use the spinner to find the probabilities indicated.\r\n\r\n\"A\r\n\r\n7. Landing on a vowel\r\n\r\n9. Landing on purple or a vowel\r\n\r\n11. Landing on green or blue\r\n\r\n13. Not landing on yellow or a consonant\r\n\r\nFor the following exercises, two coins are tossed.\r\n\r\n14. What is the sample space?\r\n\r\n15. Find the probability of tossing two heads.\r\n\r\n16.\u00a0Find the probability of tossing exactly one tail.\r\n\r\n17. Find the probability of tossing at least one tail.\r\n\r\nFor the following exercises, four coins are tossed.\r\n\r\n18. What is the sample space?\r\n\r\n19. Find the probability of tossing exactly two heads.\r\n\r\n21. Find the probability of tossing four heads or four tails.\r\n\r\n23. Find the probability of tossing not all tails.\r\n\r\nFor the following exercises, one card is drawn from a standard deck of [latex]52[\/latex] cards. Find the probability of drawing the following:\r\n\r\n27. A two\r\n\r\n29. Red six\r\n\r\n31. A non-ace\r\n\r\nFor the following exercises, two dice are rolled, and the results are summed.\r\n\r\n33. Construct a table showing the sample space of outcomes and sums.\r\n\r\n35. Find the probability of rolling at least one four or a sum of [latex]8[\/latex].\r\n\r\n37. Find the probability of rolling a sum greater than or equal to [latex]15[\/latex].\r\n\r\n39. Find the probability of rolling a sum less than [latex]6[\/latex] or greater than [latex]9[\/latex].\r\n\r\n41. Find the probability of rolling a sum of [latex]5[\/latex] or [latex]6[\/latex].\r\n\r\nFor the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following:\r\n\r\n43. A head on the coin or a club\r\n\r\n45. A head on the coin or a face card\r\n\r\nFor the following exercises, use this scenario: a bag of M&Ms contains [latex]12[\/latex] blue, [latex]6[\/latex] brown, [latex]10[\/latex] orange, [latex]8[\/latex] yellow, [latex]8[\/latex] red, and [latex]4[\/latex] green M&Ms. Reaching into the bag, a person grabs 5 M&Ms.\r\n\r\n47. What is the probability of getting all blue M&Ms?\r\n\r\n49. What is the probability of getting [latex]3[\/latex] blue M&Ms?\r\n\r\nUse this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).[footnote]United States Census Bureau. http:\/\/www.census.gov\/.[\/footnote]\r\n\r\n56. If you meet a U.S. citizen, what is the percent chance that the person is elderly? (Round to the nearest tenth of a percent.)\r\n\r\n57. If you meet five U.S. citizens, what is the percent chance that exactly one is elderly? (Round to the nearest tenth of a percent.)","rendered":"

    Counting Principles<\/h1>\n

    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?<\/p>\n

    For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.<\/p>\n

    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]<\/p>\n

    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?<\/p>\n

    11. How many outcomes are possible from tossing a coin and rolling a 6-sided die?<\/p>\n

    13. How many ways are there to construct a string of 3 digits if numbers can be repeated?<\/p>\n

    For the following exercises, compute the value of the expression.<\/p>\n

    15. [latex]P\\left(5,2\\right)[\/latex]<\/p>\n

    17. [latex]P\\left(3,3\\right)[\/latex]<\/p>\n

    21. [latex]C\\left(12,4\\right)[\/latex]<\/p>\n

    23. [latex]C\\left(7,6\\right)[\/latex]<\/p>\n

    For the following exercises, find the distinct number of arrangements.<\/p>\n

    31. The letters in the word “academia”<\/p>\n

    33. The symbols in the string #,#,#,@,@,$,$,$,%,%,%,%<\/p>\n

    35.\u00a0The 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:<\/em> use the fact that a whole number cannot start with the digit 0.)<\/p>\n

    37. Can [latex]C\\left(n,r\\right)[\/latex] ever equal [latex]P\\left(n,r\\right)?[\/latex] Explain.<\/p>\n

    41. A cell phone company offers 6 different voice packages and 8 different data packages. Of those, 3 packages include both voice and data. How many ways are there to choose either voice or data, but not both?<\/p>\n

    43. A wholesale T-shirt company offers sizes small, medium, large, and extra-large in organic or non-organic cotton and colors white, black, gray, blue, and red. How many different T-shirts are there to choose from?<\/p>\n

    45. An art store has 4 brands of paint pens in 12 different colors and 3 types of ink. How many paint pens are there to choose from?<\/p>\n

    47. How many ways can a baseball coach arrange the order of 9 batters if there are 15 players on the team?<\/p>\n

    49. A motorcycle shop has 10 choppers, 6 bobbers, and 5 caf\u00e9 racers\u2014different types of vintage motorcycles. How many ways can the shop choose 3 choppers, 5 bobbers, and 2 caf\u00e9 racers for a weekend showcase?<\/p>\n

    51. Just-For-Kicks Sneaker Company offers an online customizing service. How many ways are there to design a custom pair of Just-For-Kicks sneakers if a customer can choose from a basic shoe up to 11 customizable options?<\/p>\n

    53. Susan bought 20 plants to arrange along the border of her garden. How many distinct arrangements can she make if the plants are comprised of 6 tulips, 6 roses, and 8 daisies?<\/p>\n

    Binomial Theorem<\/h1>\n

    1. What is a binomial coefficient, and how it is calculated?<\/p>\n

    For the following exercises, evaluate the binomial coefficient.<\/p>\n

    5. [latex]\\left(\\begin{array}{c}6\\\\ 2\\end{array}\\right)[\/latex]<\/p>\n

    9. [latex]\\left(\\begin{array}{c}10\\\\ 9\\end{array}\\right)[\/latex]<\/p>\n

    For the following exercises, use the Binomial Theorem to expand each binomial.<\/p>\n

    13. [latex]{\\left(4a-b\\right)}^{3}[\/latex]<\/p>\n

    15. [latex]{\\left(3a+2b\\right)}^{3}[\/latex]<\/p>\n

    19. [latex]{\\left(4x - 3y\\right)}^{5}[\/latex]<\/p>\n

    21. [latex]{\\left({x}^{-1}+2{y}^{-1}\\right)}^{4}[\/latex]<\/p>\n

    For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.<\/p>\n

    23. [latex]{\\left(a+b\\right)}^{17}[\/latex]<\/p>\n

    25. [latex]{\\left(a - 2b\\right)}^{15}[\/latex]<\/p>\n

    27. [latex]{\\left(3a+b\\right)}^{20}[\/latex]<\/p>\n

    29. [latex]{\\left({x}^{3}-\\sqrt{y}\\right)}^{8}[\/latex]<\/p>\n

    For the following exercises, find the indicated term of each binomial without fully expanding the binomial.<\/p>\n

    33. The eighth term of [latex]{\\left(7+5y\\right)}^{14}[\/latex]<\/p>\n

    35. The fifth term of [latex]{\\left(x-y\\right)}^{7}[\/latex]<\/p>\n

    39. The eighth term of [latex]{\\left(\\frac{y}{2}+\\frac{2}{x}\\right)}^{9}[\/latex]<\/p>\n

    45. In the expansion of [latex]{\\left(5x+3y\\right)}^{n}[\/latex], each term has the form [latex]\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right){a}^{n-k}{b}^{k}, \\text{where } k[\/latex] successively takes on the value [latex]0,1,2,...,n[\/latex]. If [latex]\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right)=\\left(\\begin{array}{c}7\\\\ 2\\end{array}\\right)[\/latex], what is the corresponding term?<\/p>\n

    49.\u00a0Which expression cannot be expanded using the Binomial Theorem? Explain.<\/p>\n