{"id":1197,"date":"2026-01-19T18:52:45","date_gmt":"2026-01-19T08:52:45","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1197"},"modified":"2026-01-19T23:06:01","modified_gmt":"2026-01-19T13:06:01","slug":"year11-math-4-1-1-combinatorics","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1197","title":{"rendered":"Year11- MATH-4-1-1 Combinatorics"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">In Unit 1 of the Queensland Senior Syllabus for <strong>Specialist Mathematics<\/strong> (Grade 11), <strong>Combinatorics<\/strong> is <mark>the study of counting, arranging, and combining objects to determine the total number of possible outcomes<\/mark>. It is essentially &#8220;the mathematics of counting&#8221; used to solve problems where the number of possibilities is too large to count individually.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It acts as a foundation for probability and is used to analyze discrete, finite structures.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Key Concepts in Queensland Grade 11 Combinatorics&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">According to the QCAA curriculum (Specialist Mathematics Unit 1, Topic 1), students focus on the following core areas:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Principles of Counting:<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Addition Principle (Rule of Sum):<\/strong> Used when counting arrangements that cannot happen simultaneously (e.g., choosing either A or B).<\/li>\n\n\n\n<li><strong>Multiplication Principle (Rule of Product):<\/strong> Used for consecutive or simultaneous actions (e.g., choosing an item from category A <em>and<\/em> another from category B).<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Factorial Notation:<\/strong> Understanding \ud835\udc5b! (e.g., <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>4<\/mn><mo>!<\/mo><mo>=<\/mo><mn>4<\/mn><mo>\u00d7<\/mo><mn>3<\/mn><mo>\u00d7<\/mo><mn>2<\/mn><mo>\u00d7<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"text\/plain\">4 exclamation mark equals 4 cross 3 cross 2 cross 1<\/annotation><\/semantics><\/math>) to calculate permutations.<\/li>\n\n\n\n<li><strong>Permutations (Arrangements &#8211; Order Matters):<\/strong>\n<ul class=\"wp-block-list\">\n<li>Defining and using permutations (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><\/mrow><mi>n<\/mi><\/msup><msub><mi>P<\/mi><mi>r<\/mi><\/msub><\/mrow><annotation encoding=\"text\/plain\">to the n-th power cap P sub r<\/annotation><\/semantics><\/math>) to arrange <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>r<\/mi><annotation encoding=\"text\/plain\">r<\/annotation><\/semantics><\/math> objects from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>n<\/mi><annotation encoding=\"text\/plain\">n<\/annotation><\/semantics><\/math> distinct objects.<\/li>\n\n\n\n<li>Solving problems with restrictions, such as repeated objects or specific items grouped together.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Combinations (Selections &#8211; Order Does Not Matter):<\/strong>\n<ul class=\"wp-block-list\">\n<li>Defining and using combinations (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><\/mrow><mi>n<\/mi><\/msup><msub><mi>C<\/mi><mi>r<\/mi><\/msub><\/mrow><annotation encoding=\"text\/plain\">to the n-th power cap C sub r<\/annotation><\/semantics><\/math> or <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>(<\/mo><mfrac linethickness=\"0\"><mi>n<\/mi><mi>r<\/mi><\/mfrac><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">the 2 by 1 column matrix; n, r end-matrix;<\/annotation><\/semantics><\/math>) to choose <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>r<\/mi><annotation encoding=\"text\/plain\">r<\/annotation><\/semantics><\/math> objects from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>n<\/mi><annotation encoding=\"text\/plain\">n<\/annotation><\/semantics><\/math> distinct objects.<\/li>\n\n\n\n<li>Solving problems with restrictions, such as selecting from multiple groups.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Key Principles &amp; Theorems:<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>The Inclusion-Exclusion Principle:<\/strong> Used to find the number of elements in the union of two or three sets.<\/li>\n\n\n\n<li><strong>The Pigeonhole Principle:<\/strong> A, method for proving that if <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>n<\/mi><annotation encoding=\"text\/plain\">n<\/annotation><\/semantics><\/math> items are put into <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>m<\/mi><annotation encoding=\"text\/plain\">m<\/annotation><\/semantics><\/math> containers, with <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><mo>&gt;<\/mo><mi>m<\/mi><\/mrow><annotation encoding=\"text\/plain\">n is greater than m<\/annotation><\/semantics><\/math>, then at least one container must contain more than one item.<\/li>\n\n\n\n<li><strong>Pascal\u2019s Triangle:<\/strong> Exploring patterns in combinations.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Applications:<\/strong>\n<ul class=\"wp-block-list\">\n<li>Applying these techniques to probability problems.&nbsp;<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Examples of Problems Covered<\/strong>&nbsp;<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Arrangements:<\/strong> In how many ways can 5 people stand in a line? (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>5<\/mn><mo>!<\/mo><\/mrow><annotation encoding=\"text\/plain\">5 exclamation mark<\/annotation><\/semantics><\/math>)<\/li>\n\n\n\n<li><strong>Permutations:<\/strong> How many 4-digit PINs can be formed using 10 digits without repetition? (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><\/mrow><mn>10<\/mn><\/msup><msub><mi>P<\/mi><mn>4<\/mn><\/msub><\/mrow><annotation encoding=\"text\/plain\">to the tenth power cap P sub 4<\/annotation><\/semantics><\/math>)<\/li>\n\n\n\n<li><strong>Combinations:<\/strong> How many ways can a committee of 3 be chosen from 10 people? (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><\/mrow><mn>10<\/mn><\/msup><msub><mi>C<\/mi><mn>3<\/mn><\/msub><\/mrow><annotation encoding=\"text\/plain\">to the tenth power cap C sub 3<\/annotation><\/semantics><\/math>)<\/li>\n\n\n\n<li><strong>Restrictions:<\/strong> How many ways can 5 books be arranged if 2 specific books must be together?&nbsp;<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 1: Grouping the Restricted Items&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Treat the 2 specific books that must be together as a single unit or &#8220;block.&#8221; This reduces the total number of items to arrange from 5 individual books to 4 items (the 1 block containing the 2 specific books plus the remaining 3 individual books).&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 2: Arranging the Items&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate the number of ways to arrange these 4 items. The number of permutations of <math data-latex=\"n\"><semantics><mi>n<\/mi><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math> distinct items is given by <math data-latex=\"n!\"><semantics><mrow><mi>n<\/mi><mo form=\"postfix\" stretchy=\"false\">!<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">n!<\/annotation><\/semantics><\/math>. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math data-latex=\"4!=4\\times 3\\times 2\\times 1=24\"><semantics><mrow><mn>4<\/mn><mo form=\"postfix\" stretchy=\"false\">!<\/mo><mo>=<\/mo><mn>4<\/mn><mo>\u00d7<\/mo><mn>3<\/mn><mo>\u00d7<\/mo><mn>2<\/mn><mo>\u00d7<\/mo><mn>1<\/mn><mo>=<\/mo><mn>24<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4!=4\\times 3\\times 2\\times 1=24<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 3: Accounting for Internal Permutations&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Within the block of 2 specific books, the books can be arranged among themselves. The number of ways to arrange 2 books is <math data-latex=\"2!\"><semantics><mrow><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">!<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">2!<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"> <math data-latex=\"2!=2\\times 1=2\"><semantics><mrow><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">!<\/mo><mo>=<\/mo><mn>2<\/mn><mo>\u00d7<\/mo><mn>1<\/mn><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2!=2\\times 1=2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 4: Calculating Total Arrangements&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Multiply the number of ways to arrange the 4 items by the number of internal arrangements of the specific books to find the total permutations.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"> <math data-latex=\"24\\times 2=48\"><semantics><mrow><mn>24<\/mn><mo>\u00d7<\/mo><mn>2<\/mn><mo>=<\/mo><mn>48<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">24\\times 2=48<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&nbsp;<strong>Answer:<\/strong>&nbsp;      The 5 books can be arranged in <strong>48<\/strong> different ways.&nbsp;<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\">Combinatorics in this unit often utilizes technology to solve complex problems.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">***************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30af\u30a4\u30fc\u30f3\u30ba\u30e9\u30f3\u30c9\u5dde\u5c02\u9580\u6570\u5b66\uff0811\u5e74\u751f\uff09\u306e\u30b7\u30cb\u30a2\u30b7\u30e9\u30d0\u30b9\u306e\u30e6\u30cb\u30c3\u30c81\u3067\u306f\u3001\u7d44\u5408\u305b\u8ad6\u3068\u306f\u3001\u7269\u4f53\u3092\u6570\u3048\u3001\u4e26\u3079\u3001\u7d44\u307f\u5408\u308f\u305b\u3066\u3001\u8d77\u3053\u308a\u5f97\u308b\u7d50\u679c\u306e\u7dcf\u6570\u3092\u6c7a\u5b9a\u3059\u308b\u5b66\u554f\u3067\u3059\u3002\u3053\u308c\u306f\u672c\u8cea\u7684\u306b\u300c\u6570\u3048\u308b\u6570\u5b66\u300d\u3067\u3042\u308a\u3001\u53ef\u80fd\u6027\u306e\u6570\u304c\u591a\u3059\u304e\u3066\u500b\u5225\u306b\u6570\u3048\u3089\u308c\u306a\u3044\u554f\u984c\u3092\u89e3\u304f\u305f\u3081\u306b\u7528\u3044\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7d44\u5408\u305b\u8ad6\u306f\u78ba\u7387\u306e\u57fa\u790e\u3068\u3057\u3066\u6a5f\u80fd\u3057\u3001\u96e2\u6563\u7684\u3067\u6709\u9650\u306a\u69cb\u9020\u3092\u5206\u6790\u3059\u308b\u305f\u3081\u306b\u7528\u3044\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30af\u30a4\u30fc\u30f3\u30ba\u30e9\u30f3\u30c9\u5dde11\u5e74\u751f \u7d44\u307f\u5408\u308f\u305b\u8ad6\u306e\u4e3b\u8981\u6982\u5ff5<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">QCAA\u30ab\u30ea\u30ad\u30e5\u30e9\u30e0\uff08\u5c02\u9580\u6570\u5b66\u30e6\u30cb\u30c3\u30c81\u3001\u30c8\u30d4\u30c3\u30af1\uff09\u306b\u57fa\u3065\u304d\u3001\u751f\u5f92\u306f\u4ee5\u4e0b\u306e\u4e2d\u6838\u5206\u91ce\u306b\u91cd\u70b9\u7684\u306b\u53d6\u308a\u7d44\u307f\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6570\u3048\u65b9\u306e\u539f\u5247\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u52a0\u6cd5\u306e\u539f\u5247\uff08\u548c\u306e\u6cd5\u5247\uff09\uff1a\u540c\u6642\u306b\u8d77\u3053\u308a\u5f97\u306a\u3044\u7d44\u307f\u5408\u308f\u305b\u3092\u6570\u3048\u308b\u969b\u306b\u7528\u3044\u3089\u308c\u307e\u3059\uff08\u4f8b\uff1aA\u307e\u305f\u306fB\u306e\u3044\u305a\u308c\u304b\u3092\u9078\u629e\u3059\u308b\uff09\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e57\u6cd5\u306e\u539f\u5247\uff08\u7a4d\u306e\u6cd5\u5247\uff09\uff1a\u9023\u7d9a\u3057\u305f\u3001\u307e\u305f\u306f\u540c\u6642\u306b\u8d77\u3053\u308b\u52d5\u4f5c\u306b\u7528\u3044\u3089\u308c\u307e\u3059\uff08\u4f8b\uff1a\u30ab\u30c6\u30b4\u30ea\u30fcA\u304b\u30891\u3064\u306e\u9805\u76ee\u3092\u9078\u629e\u3057\u3001\u30ab\u30c6\u30b4\u30ea\u30fcB\u304b\u3089\u5225\u306e\u9805\u76ee\u3092\u9078\u629e\u3059\u308b\uff09\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u968e\u4e57\u306e\u8868\u8a18\u6cd5\uff1a\u9806\u5217\u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306b\u3001<math data-latex=\"n!\"><semantics><mrow><mi>n<\/mi><mo form=\"postfix\" stretchy=\"false\">!<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">n!<\/annotation><\/semantics><\/math>\uff08\u4f8b\uff1a<math data-latex=\"4!=4\\times 3\\times 2\\times 1\"><semantics><mrow><mn>4<\/mn><mo form=\"postfix\" stretchy=\"false\">!<\/mo><mo>=<\/mo><mn>4<\/mn><mo>\u00d7<\/mo><mn>3<\/mn><mo>\u00d7<\/mo><mn>2<\/mn><mo>\u00d7<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4!=4\\times 3\\times 2\\times 1<\/annotation><\/semantics><\/math>\uff09\u3092\u7406\u89e3\u3059\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u9806\u5217\uff08\u914d\u7f6e &#8211; \u9806\u5e8f\u304c\u91cd\u8981\uff09\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u9806\u5217<\/strong>\uff08<math data-latex=\"{}^{n}P_{r}\"><semantics><mrow><msup><mrow><\/mrow><mi>n<\/mi><\/msup><msub><mi>P<\/mi><mi>r<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">{}^{n}P_{r}<\/annotation><\/semantics><\/math>)\u3092\u5b9a\u7fa9\u3057\u3066\u4f7f\u7528\u3057\u3001<math data-latex=\"n\"><semantics><mi>n<\/mi><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math> \u500b\u306e\u7570\u306a\u308b\u30aa\u30d6\u30b8\u30a7\u30af\u30c8\u304b\u3089 <math data-latex=\"r\"><semantics><mi>r<\/mi><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math> \u500b\u306e\u30aa\u30d6\u30b8\u30a7\u30af\u30c8\u3092\u914d\u7f6e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u91cd\u8907\u3059\u308b\u30aa\u30d6\u30b8\u30a7\u30af\u30c8\u3084\u7279\u5b9a\u306e\u30a2\u30a4\u30c6\u30e0\u3092\u30b0\u30eb\u30fc\u30d7\u5316\u3059\u308b\u306a\u3069\u3001\u5236\u7d04\u306e\u3042\u308b\u554f\u984c\u3092\u89e3\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u7d44\u307f\u5408\u308f\u305b<\/strong>\uff08\u9078\u629e &#8211; \u9806\u5e8f\u306f\u91cd\u8981\u3067\u306f\u3042\u308a\u307e\u305b\u3093\uff09\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7d44\u307f\u5408\u308f\u305b\uff08<math data-latex=\"{}^{n}C_{r}\"><semantics><mrow><msup><mrow><\/mrow><mi>n<\/mi><\/msup><msub><mi>C<\/mi><mi>r<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">{}^{n}C_{r}<\/annotation><\/semantics><\/math> \u307e\u305f\u306f <math data-latex=\"{n \\choose r}\"><semantics><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mi>n<\/mi><mi>r<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">{n \\choose r}<\/annotation><\/semantics><\/math>\uff09\u3092\u5b9a\u7fa9\u3057\u3066\u4f7f\u7528\u3057\u3001n \u500b\u306e\u7570\u306a\u308b\u30aa\u30d6\u30b8\u30a7\u30af\u30c8\u304b\u3089 r \u500b\u306e\u30aa\u30d6\u30b8\u30a7\u30af\u30c8\u3092\u9078\u629e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u8907\u6570\u306e\u30b0\u30eb\u30fc\u30d7\u304b\u3089\u9078\u629e\u3059\u308b\u306a\u3069\u3001\u5236\u7d04\u306e\u3042\u308b\u554f\u984c\u3092\u89e3\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u4e3b\u8981\u306a\u539f\u7406\u3068\u5b9a\u7406<\/strong>\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u5305\u542b\u6392\u4ed6\u539f\u7406<\/strong>\uff1a2\u3064\u307e\u305f\u306f3\u3064\u306e\u96c6\u5408\u306e\u548c\u96c6\u5408\u306e\u8981\u7d20\u6570\u3092\u6c42\u3081\u308b\u969b\u306b\u7528\u3044\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u9ce9\u306e\u5de3\u539f\u7406<\/strong>\uff1an \u500b\u306e\u8981\u7d20\u3092m \u500b\u306e\u5bb9\u5668\uff08n&gt;m\uff09\u306b\u5165\u308c\u308b\u5834\u5408\u3001\u5c11\u306a\u304f\u3068\u30821\u3064\u306e\u5bb9\u5668\u306b\u306f\u8907\u6570\u306e\u8981\u7d20\u304c\u542b\u307e\u308c\u308b\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b\u65b9\u6cd5\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u30d1\u30b9\u30ab\u30eb\u306e\u4e09\u89d2\u5f62<\/strong>\uff1a\u7d44\u307f\u5408\u308f\u305b\u306e\u30d1\u30bf\u30fc\u30f3\u3092\u63a2\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u5fdc\u7528<\/strong>\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u308c\u3089\u306e\u624b\u6cd5\u3092\u78ba\u7387\u554f\u984c\u306b\u9069\u7528\u3059\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u6271\u3046\u554f\u984c\u306e\u4f8b<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u914d\u7f6e<\/strong>\uff1a5\u4eba\u304c\u4e00\u5217\u306b\u4e26\u3076\u65b9\u6cd5\u306f\u4f55\u901a\u308a\u3042\u308b\u304b\uff1f\uff08<math data-latex=\"5!\"><semantics><mrow><mn>5<\/mn><mo form=\"postfix\" stretchy=\"false\">!<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">5!<\/annotation><\/semantics><\/math>\uff09<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u9806\u5217<\/strong>\uff1a10\u6841\u306e\u6570\u5b57\u3092\u91cd\u8907\u306a\u304f\u7d44\u307f\u5408\u308f\u305b\u3066\u30014\u6841\u306e\u6697\u8a3c\u756a\u53f7\u3092\u3044\u304f\u3064\u4f5c\u6210\u3067\u304d\u308b\u304b\uff1f\uff08<math data-latex=\"{}^{10}P_{4}\"><semantics><mrow><msup><mrow><\/mrow><mn>10<\/mn><\/msup><msub><mi>P<\/mi><mn>4<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">{}^{10}P_{4}<\/annotation><\/semantics><\/math>\uff09<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u7d44\u307f\u5408\u308f\u305b<\/strong>\uff1a10\u4eba\u306e\u4e2d\u304b\u30893\u4eba\u3067\u69cb\u6210\u3055\u308c\u308b\u59d4\u54e1\u4f1a\u3092\u9078\u51fa\u3059\u308b\u65b9\u6cd5\u306f\u4f55\u901a\u308a\u3042\u308b\u304b\uff1f (<math data-latex=\"{}^{10}C_{3}\"><semantics><mrow><msup><mrow><\/mrow><mn>10<\/mn><\/msup><msub><mi>C<\/mi><mn>3<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">{}^{10}C_{3}<\/annotation><\/semantics><\/math>)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u5236\u7d04<\/strong>\uff1a\u7279\u5b9a\u306e2\u518a\u306e\u672c\u3092\u5fc5\u305a\u4e00\u7dd2\u306b\u4e26\u3079\u308b\u5834\u5408\u30015\u518a\u306e\u672c\u3092\u4e26\u3079\u308b\u65b9\u6cd5\u306f\u4f55\u901a\u308a\u3042\u308a\u307e\u3059\u304b\uff1f<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30b9\u30c6\u30c3\u30d71\uff1a\u5236\u9650\u3055\u308c\u305f\u30a2\u30a4\u30c6\u30e0\u306e\u30b0\u30eb\u30fc\u30d7\u5316\u3000\u4e00\u7dd2\u306b\u7f6e\u304b\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u7279\u5b9a\u306e2\u518a\u306e\u672c\u3092\u30011\u3064\u306e\u30e6\u30cb\u30c3\u30c8\u3001\u3064\u307e\u308a\u300c\u30d6\u30ed\u30c3\u30af\u300d\u3068\u3057\u3066\u6271\u3044\u307e\u3059\u3002<br>\u3053\u308c\u306b\u3088\u308a\u3001\u914d\u7f6e\u3059\u308b\u30a2\u30a4\u30c6\u30e0\u306e\u7dcf\u6570\u304c5\u518a\u306e\u672c\u304b\u30894\u518a\uff08\u7279\u5b9a\u306e2\u518a\u306e\u672c\u3092\u542b\u30801\u3064\u306e\u30d6\u30ed\u30c3\u30af\u3068\u3001\u6b8b\u308a\u306e3\u518a\u306e\u672c\uff09\u306b\u6e1b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30b9\u30c6\u30c3\u30d72\uff1a\u30a2\u30a4\u30c6\u30e0\u306e\u914d\u7f6e\u3000\u3053\u308c\u3089\u306e4\u3064\u306e\u30a2\u30a4\u30c6\u30e0\u3092\u914d\u7f6e\u3059\u308b\u65b9\u6cd5\u306e\u6570\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002(<math data-latex=\"n\"><semantics><mi>n<\/mi><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math>)\u500b\u306e\u7570\u306a\u308b\u30a2\u30a4\u30c6\u30e0\u306e\u9806\u5217\u306e\u6570\u306f(<math data-latex=\"n!\"><semantics><mrow><mi>n<\/mi><mo form=\"postfix\" stretchy=\"false\">!<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">n!<\/annotation><\/semantics><\/math>).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math data-latex=\"(4!=4\\times 3\\times 2\\times 1=24)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>4<\/mn><mo form=\"postfix\" stretchy=\"false\">!<\/mo><mo>=<\/mo><mn>4<\/mn><mo>\u00d7<\/mo><mn>3<\/mn><mo>\u00d7<\/mo><mn>2<\/mn><mo>\u00d7<\/mo><mn>1<\/mn><mo>=<\/mo><mn>24<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(4!=4\\times 3\\times 2\\times 1=24)<\/annotation><\/semantics><\/math>\u3067\u4e0e\u3048\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30b9\u30c6\u30c3\u30d73\uff1a\u5185\u90e8\u9806\u5217\u306e\u8003\u616e\u3000\u7279\u5b9a\u306e2\u518a\u306e\u672c\u306e\u30d6\u30ed\u30c3\u30af\u5185\u3067\u306f\u3001\u672c\u540c\u58eb\u3092\u4e26\u3079\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002 2\u518a\u306e\u672c\u306e\u4e26\u3079\u65b9\u306f2\u901a\u308a\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(<math data-latex=\"2!=2\\times 1=2\"><semantics><mrow><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">!<\/mo><mo>=<\/mo><mn>2<\/mn><mo>\u00d7<\/mo><mn>1<\/mn><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2!=2\\times 1=2<\/annotation><\/semantics><\/math>)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30b9\u30c6\u30c3\u30d74\uff1a\u5408\u8a08\u306e\u4e26\u3073\u65b9\u3092\u8a08\u7b97\u3059\u308b\u30004\u3064\u306e\u30a2\u30a4\u30c6\u30e0\u306e\u4e26\u3079\u65b9\u3068\u3001\u305d\u308c\u305e\u308c\u306e\u672c\u306e\u5185\u90e8\u306e\u4e26\u3073\u65b9\u3092\u639b\u3051\u3066\u3001\u5408\u8a08\u306e\u4e26\u3073\u65b9\u3092\u6c42\u3081\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(<math data-latex=\"24\\times 2=48\"><semantics><mrow><mn>24<\/mn><mo>\u00d7<\/mo><mn>2<\/mn><mo>=<\/mo><mn>48<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">24\\times 2=48<\/annotation><\/semantics><\/math>)      \u7b54\u3048\uff1a5\u518a\u306e\u672c\u306f48\u901a\u308a\u306e\u4e26\u3073\u65b9\u304c\u53ef\u80fd\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>>> \u3053\u306e\u5358\u5143\u306e\u7d44\u5408\u305b\u8ad6\u3067\u306f\u3001\u8907\u96d1\u306a\u554f\u984c\u3092\u89e3\u6c7a\u3059\u308b\u305f\u3081\u306b\u30c6\u30af\u30ce\u30ed\u30b8\u30fc\u3092\u3088\u304f\u5229\u7528\u3057\u307e\u3059\u3002<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In Unit 1 of the Queensland Senior Syllabus for Specialist Mathematics (Grade 11), Combinatorics is the study of counting, arranging, and combining objects to determine the total number of possible outcomes. It is essentially &#8220;the mathematics of counting&#8221; used to solve problems where the number of possibilities is too large to count individually.&nbsp; It acts [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1197","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1197","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1197"}],"version-history":[{"count":6,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1197\/revisions"}],"predecessor-version":[{"id":1221,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1197\/revisions\/1221"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1197"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1197"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1197"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}