{"id":146,"date":"2026-01-12T15:50:41","date_gmt":"2026-01-12T15:50:41","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&p=146"},"modified":"2026-01-12T15:50:42","modified_gmt":"2026-01-12T15:50:42","slug":"probability-and-counting-principles-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/probability-and-counting-principles-get-stronger-answer-key\/","title":{"raw":"Probability and Counting Principles: Get Stronger Answer Key","rendered":"Probability and Counting Principles: Get Stronger Answer Key"},"content":{"raw":"

Counting Principles<\/h2>\r\n
    \r\n \t
  1. [latex]4 + 2 = 6[\/latex]<\/li>\r\n \t
  2. [latex]5 + 4 + 7 = 16[\/latex]<\/li>\r\n \t
  3. [latex]2 \\times 6 = 12[\/latex]<\/li>\r\n \t
  4. [latex]10^3 = 1000[\/latex]<\/li>\r\n \t
  5. [latex]P(5,2) = 20[\/latex]<\/li>\r\n \t
  6. [latex]P(3,3) = 6[\/latex]<\/li>\r\n \t
  7. [latex]P(11,5) = 55,440[\/latex]<\/li>\r\n \t
  8. [latex]C(12,4) = 495[\/latex]<\/li>\r\n \t
  9. [latex]C(7,6) = 7[\/latex]<\/li>\r\n \t
  10. [latex]2^{10} = 1024[\/latex]<\/li>\r\n \t
  11. [latex]2^{12} = 4096[\/latex]<\/li>\r\n \t
  12. [latex]2^9 = 512[\/latex]<\/li>\r\n \t
  13. [latex]\\frac{8!}{3!} = 6720[\/latex]<\/li>\r\n \t
  14. [latex]\\frac{12!}{3!2!3!4!}[\/latex]<\/li>\r\n \t
  15. [latex]6 - 3 + 8 - 3 = 8[\/latex]<\/li>\r\n \t
  16. [latex]4 \\times 2 \\times 5 = 40[\/latex]<\/li>\r\n \t
  17. [latex]4 \\times 12 \\times 3 = 144[\/latex]<\/li>\r\n \t
  18. [latex]P(15,9) = 1,816,214,400[\/latex]<\/li>\r\n \t
  19. [latex]C(10,3) \\times C(6,5) \\times C(5,2) = 7,200[\/latex]<\/li>\r\n \t
  20. [latex]2^{11} = 2048[\/latex]<\/li>\r\n \t
  21. [latex]\\frac{20!}{6!6!8!} = 116,396,280[\/latex]<\/li>\r\n<\/ol>\r\n

    Binomial Theorem<\/h2>\r\n
      \r\n \t
    1. [latex]15[\/latex]<\/li>\r\n \t
    2. [latex]35[\/latex]<\/li>\r\n \t
    3. [latex]10[\/latex]<\/li>\r\n \t
    4. [latex]12,376[\/latex]<\/li>\r\n \t
    5. [latex]64a^3 - 48a^2b + 12ab^2 - b^3[\/latex]<\/li>\r\n \t
    6. [latex]27a^3 + 54a^2b + 36ab^2 + 8b^3[\/latex]<\/li>\r\n \t
    7. [latex]1024x^5 + 2560x^4y + 2560x^3y^2 + 128x^2y^3 + 320xy^4 + 32y^5[\/latex]<\/li>\r\n \t
    8. [latex]1024x^5 - 3840x^4y + 5760x^3y^2 - 4320x^2y^3 + 1620xy^4 - 243y^5[\/latex]<\/li>\r\n \t
    9. [latex]\\frac{1}{x} + \\frac{24}{x^5y} + \\frac{32}{x^8y^2} + \\frac{16}{x^{11}y^3} + \\frac{3}{y^4}[\/latex]<\/li>\r\n \t
    10. [latex]a^{17} + 17a^{16}b + 136a^{15}b^2[\/latex]<\/li>\r\n \t
    11. [latex]a^{15} - 30a^{14}b + 420a^{13}b^2[\/latex]<\/li>\r\n \t
    12. [latex]3,486,784,401a^{20} + 23,245,229,340a^{19}b + 73,609,892,910a^{18}b^2[\/latex]<\/li>\r\n \t
    13. [latex]x^{24} - 8x^{21}\\sqrt{y} + 28x^{18}y[\/latex]<\/li>\r\n \t
    14. [latex]-720x^2y^3[\/latex]<\/li>\r\n \t
    15. [latex]220,812,466,875,000y^7[\/latex]<\/li>\r\n \t
    16. [latex]35x^3y^4[\/latex]<\/li>\r\n \t
    17. [latex]1,082,565a^3b^{10}[\/latex]<\/li>\r\n \t
    18. [latex]\\frac{1152y^2}{x^7}[\/latex]<\/li>\r\n<\/ol>\r\n

      Introduction to Probability<\/h2>\r\n
        \r\n \t
      1. [latex]\\frac{1}{2}[\/latex]<\/li>\r\n \t
      2. [latex]\\frac{5}{8}[\/latex]<\/li>\r\n \t
      3. [latex]\\frac{1}{2}[\/latex]<\/li>\r\n \t
      4. [latex]\\frac{3}{8}[\/latex]<\/li>\r\n \t
      5. [latex]\\frac{1}{4}[\/latex]<\/li>\r\n \t
      6. [latex]\\frac{3}{4}[\/latex]<\/li>\r\n \t
      7. [latex]\\frac{3}{8}[\/latex]<\/li>\r\n \t
      8. [latex]\\frac{1}{8}[\/latex]<\/li>\r\n \t
      9. [latex]\\frac{15}{16}[\/latex]<\/li>\r\n \t
      10. [latex]\\frac{5}{8}[\/latex]<\/li>\r\n \t
      11. [latex]\\frac{1}{13}[\/latex]<\/li>\r\n \t
      12. [latex]\\frac{1}{26}[\/latex]<\/li>\r\n \t
      13. [latex]\\frac{12}{13}[\/latex]<\/li>\r\n \t
      14. \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
        <\/th>\r\n1<\/th>\r\n2<\/th>\r\n3<\/th>\r\n4<\/th>\r\n5<\/th>\r\n6<\/th>\r\n<\/tr>\r\n
        1<\/td>\r\n[latex](1,1)\\\\2[\/latex]<\/td>\r\n[latex](1,2)\\\\3[\/latex]<\/td>\r\n[latex](1,3)\\\\4[\/latex]<\/td>\r\n[latex](1,4)\\\\5[\/latex]<\/td>\r\n[latex](1,5)\\\\6[\/latex]<\/td>\r\n[latex](1,6)\\\\7[\/latex]<\/td>\r\n<\/tr>\r\n
        2<\/td>\r\n[latex](2,1)\\\\3[\/latex]<\/td>\r\n[latex](2,2)\\\\4[\/latex]<\/td>\r\n[latex](2,3)\\\\5[\/latex]<\/td>\r\n[latex](2,4)\\\\6[\/latex]<\/td>\r\n[latex](2,5)\\\\7[\/latex]<\/td>\r\n[latex](2,6)\\\\8[\/latex]<\/td>\r\n<\/tr>\r\n
        3<\/td>\r\n[latex](3,1)\\\\4[\/latex]<\/td>\r\n[latex](3,2)\\\\5[\/latex]<\/td>\r\n[latex](3,3)\\\\6[\/latex]<\/td>\r\n[latex](3,4)\\\\7[\/latex]<\/td>\r\n[latex](3,5)\\\\8[\/latex]<\/td>\r\n[latex](3,6)\\\\9[\/latex]<\/td>\r\n<\/tr>\r\n
        4<\/td>\r\n[latex](4,1)\\\\5[\/latex]<\/td>\r\n[latex](4,2)\\\\6[\/latex]<\/td>\r\n[latex](4,3)\\\\7[\/latex]<\/td>\r\n[latex](4,4)\\\\8[\/latex]<\/td>\r\n[latex](4,5)\\\\9[\/latex]<\/td>\r\n[latex](4,6)\\\\10[\/latex]<\/td>\r\n<\/tr>\r\n
        5<\/td>\r\n[latex](5,1)\\\\6[\/latex]<\/td>\r\n[latex](5,2)\\\\7[\/latex]<\/td>\r\n[latex](5,3)\\\\8[\/latex]<\/td>\r\n[latex](5,4)\\\\9[\/latex]<\/td>\r\n[latex](5,5)\\\\10[\/latex]<\/td>\r\n[latex](5,6)\\\\11[\/latex]<\/td>\r\n<\/tr>\r\n
        6<\/td>\r\n[latex](6,1)\\\\7[\/latex]<\/td>\r\n[latex](6,2)\\\\8[\/latex]<\/td>\r\n[latex](6,3)\\\\9[\/latex]<\/td>\r\n[latex](6,4)\\\\10[\/latex]<\/td>\r\n[latex](6,5)\\\\11[\/latex]<\/td>\r\n[latex](6,6)\\\\12[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n
          \r\n \t
        1. [latex]\\frac{5}{12}[\/latex]<\/li>\r\n \t
        2. [latex]0[\/latex]<\/li>\r\n \t
        3. [latex]\\frac{4}{9}[\/latex]<\/li>\r\n \t
        4. [latex]\\frac{1}{4}[\/latex]<\/li>\r\n \t
        5. [latex]\\frac{5}{8}[\/latex]<\/li>\r\n \t
        6. [latex]\\frac{3}{13}[\/latex]<\/li>\r\n \t
        7. [latex]\\frac{C(12,5)}{C(48,5)} = \\frac{1}{2162}[\/latex]<\/li>\r\n \t
        8. [latex]\\frac{C(12,3)C(36,2)}{C(48,5)} = \\frac{175}{2162}[\/latex]<\/li>\r\n \t
        9. [latex]\\frac{C(55800000,1)C(275200000,4)}{C(331000000,5)} = 40.28 \\%[\/latex]<\/li>\r\n \t
        10. [latex]\\frac{C(55800000,4)C(275200000,1)}{C(331000000,5)} = .336 \\%[\/latex]<\/li>\r\n<\/ol>","rendered":"

          Counting Principles<\/h2>\n
            \n
          1. [latex]4 + 2 = 6[\/latex]<\/li>\n
          2. [latex]5 + 4 + 7 = 16[\/latex]<\/li>\n
          3. [latex]2 \\times 6 = 12[\/latex]<\/li>\n
          4. [latex]10^3 = 1000[\/latex]<\/li>\n
          5. [latex]P(5,2) = 20[\/latex]<\/li>\n
          6. [latex]P(3,3) = 6[\/latex]<\/li>\n
          7. [latex]P(11,5) = 55,440[\/latex]<\/li>\n
          8. [latex]C(12,4) = 495[\/latex]<\/li>\n
          9. [latex]C(7,6) = 7[\/latex]<\/li>\n
          10. [latex]2^{10} = 1024[\/latex]<\/li>\n
          11. [latex]2^{12} = 4096[\/latex]<\/li>\n
          12. [latex]2^9 = 512[\/latex]<\/li>\n
          13. [latex]\\frac{8!}{3!} = 6720[\/latex]<\/li>\n
          14. [latex]\\frac{12!}{3!2!3!4!}[\/latex]<\/li>\n
          15. [latex]6 - 3 + 8 - 3 = 8[\/latex]<\/li>\n
          16. [latex]4 \\times 2 \\times 5 = 40[\/latex]<\/li>\n
          17. [latex]4 \\times 12 \\times 3 = 144[\/latex]<\/li>\n
          18. [latex]P(15,9) = 1,816,214,400[\/latex]<\/li>\n
          19. [latex]C(10,3) \\times C(6,5) \\times C(5,2) = 7,200[\/latex]<\/li>\n
          20. [latex]2^{11} = 2048[\/latex]<\/li>\n
          21. [latex]\\frac{20!}{6!6!8!} = 116,396,280[\/latex]<\/li>\n<\/ol>\n

            Binomial Theorem<\/h2>\n
              \n
            1. [latex]15[\/latex]<\/li>\n
            2. [latex]35[\/latex]<\/li>\n
            3. [latex]10[\/latex]<\/li>\n
            4. [latex]12,376[\/latex]<\/li>\n
            5. [latex]64a^3 - 48a^2b + 12ab^2 - b^3[\/latex]<\/li>\n
            6. [latex]27a^3 + 54a^2b + 36ab^2 + 8b^3[\/latex]<\/li>\n
            7. [latex]1024x^5 + 2560x^4y + 2560x^3y^2 + 128x^2y^3 + 320xy^4 + 32y^5[\/latex]<\/li>\n
            8. [latex]1024x^5 - 3840x^4y + 5760x^3y^2 - 4320x^2y^3 + 1620xy^4 - 243y^5[\/latex]<\/li>\n
            9. [latex]\\frac{1}{x} + \\frac{24}{x^5y} + \\frac{32}{x^8y^2} + \\frac{16}{x^{11}y^3} + \\frac{3}{y^4}[\/latex]<\/li>\n
            10. [latex]a^{17} + 17a^{16}b + 136a^{15}b^2[\/latex]<\/li>\n
            11. [latex]a^{15} - 30a^{14}b + 420a^{13}b^2[\/latex]<\/li>\n
            12. [latex]3,486,784,401a^{20} + 23,245,229,340a^{19}b + 73,609,892,910a^{18}b^2[\/latex]<\/li>\n
            13. [latex]x^{24} - 8x^{21}\\sqrt{y} + 28x^{18}y[\/latex]<\/li>\n
            14. [latex]-720x^2y^3[\/latex]<\/li>\n
            15. [latex]220,812,466,875,000y^7[\/latex]<\/li>\n
            16. [latex]35x^3y^4[\/latex]<\/li>\n
            17. [latex]1,082,565a^3b^{10}[\/latex]<\/li>\n
            18. [latex]\\frac{1152y^2}{x^7}[\/latex]<\/li>\n<\/ol>\n

              Introduction to Probability<\/h2>\n
                \n
              1. [latex]\\frac{1}{2}[\/latex]<\/li>\n
              2. [latex]\\frac{5}{8}[\/latex]<\/li>\n
              3. [latex]\\frac{1}{2}[\/latex]<\/li>\n
              4. [latex]\\frac{3}{8}[\/latex]<\/li>\n
              5. [latex]\\frac{1}{4}[\/latex]<\/li>\n
              6. [latex]\\frac{3}{4}[\/latex]<\/li>\n
              7. [latex]\\frac{3}{8}[\/latex]<\/li>\n
              8. [latex]\\frac{1}{8}[\/latex]<\/li>\n
              9. [latex]\\frac{15}{16}[\/latex]<\/li>\n
              10. [latex]\\frac{5}{8}[\/latex]<\/li>\n
              11. [latex]\\frac{1}{13}[\/latex]<\/li>\n
              12. [latex]\\frac{1}{26}[\/latex]<\/li>\n
              13. [latex]\\frac{12}{13}[\/latex]<\/li>\n
              14. \n\n\n\n\n\n\n\n\n\n
                <\/th>\n1<\/th>\n2<\/th>\n3<\/th>\n4<\/th>\n5<\/th>\n6<\/th>\n<\/tr>\n
                1<\/td>\n[latex](1,1)\\\\2[\/latex]<\/td>\n[latex](1,2)\\\\3[\/latex]<\/td>\n[latex](1,3)\\\\4[\/latex]<\/td>\n[latex](1,4)\\\\5[\/latex]<\/td>\n[latex](1,5)\\\\6[\/latex]<\/td>\n[latex](1,6)\\\\7[\/latex]<\/td>\n<\/tr>\n
                2<\/td>\n[latex](2,1)\\\\3[\/latex]<\/td>\n[latex](2,2)\\\\4[\/latex]<\/td>\n[latex](2,3)\\\\5[\/latex]<\/td>\n[latex](2,4)\\\\6[\/latex]<\/td>\n[latex](2,5)\\\\7[\/latex]<\/td>\n[latex](2,6)\\\\8[\/latex]<\/td>\n<\/tr>\n
                3<\/td>\n[latex](3,1)\\\\4[\/latex]<\/td>\n[latex](3,2)\\\\5[\/latex]<\/td>\n[latex](3,3)\\\\6[\/latex]<\/td>\n[latex](3,4)\\\\7[\/latex]<\/td>\n[latex](3,5)\\\\8[\/latex]<\/td>\n[latex](3,6)\\\\9[\/latex]<\/td>\n<\/tr>\n
                4<\/td>\n[latex](4,1)\\\\5[\/latex]<\/td>\n[latex](4,2)\\\\6[\/latex]<\/td>\n[latex](4,3)\\\\7[\/latex]<\/td>\n[latex](4,4)\\\\8[\/latex]<\/td>\n[latex](4,5)\\\\9[\/latex]<\/td>\n[latex](4,6)\\\\10[\/latex]<\/td>\n<\/tr>\n
                5<\/td>\n[latex](5,1)\\\\6[\/latex]<\/td>\n[latex](5,2)\\\\7[\/latex]<\/td>\n[latex](5,3)\\\\8[\/latex]<\/td>\n[latex](5,4)\\\\9[\/latex]<\/td>\n[latex](5,5)\\\\10[\/latex]<\/td>\n[latex](5,6)\\\\11[\/latex]<\/td>\n<\/tr>\n
                6<\/td>\n[latex](6,1)\\\\7[\/latex]<\/td>\n[latex](6,2)\\\\8[\/latex]<\/td>\n[latex](6,3)\\\\9[\/latex]<\/td>\n[latex](6,4)\\\\10[\/latex]<\/td>\n[latex](6,5)\\\\11[\/latex]<\/td>\n[latex](6,6)\\\\12[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n
                  \n
                1. [latex]\\frac{5}{12}[\/latex]<\/li>\n
                2. [latex]0[\/latex]<\/li>\n
                3. [latex]\\frac{4}{9}[\/latex]<\/li>\n
                4. [latex]\\frac{1}{4}[\/latex]<\/li>\n
                5. [latex]\\frac{5}{8}[\/latex]<\/li>\n
                6. [latex]\\frac{3}{13}[\/latex]<\/li>\n
                7. [latex]\\frac{C(12,5)}{C(48,5)} = \\frac{1}{2162}[\/latex]<\/li>\n
                8. [latex]\\frac{C(12,3)C(36,2)}{C(48,5)} = \\frac{175}{2162}[\/latex]<\/li>\n
                9. [latex]\\frac{C(55800000,1)C(275200000,4)}{C(331000000,5)} = 40.28 \\%[\/latex]<\/li>\n
                10. [latex]\\frac{C(55800000,4)C(275200000,1)}{C(331000000,5)} = .336 \\%[\/latex]<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":17,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":106,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/146"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/146\/revisions"}],"predecessor-version":[{"id":147,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/146\/revisions\/147"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/106"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/146\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=146"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=146"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=146"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=146"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}