{"id":16645,"date":"2025-06-11T17:05:16","date_gmt":"2025-06-11T17:05:16","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=16645"},"modified":"2025-06-11T17:05:20","modified_gmt":"2025-06-11T17:05:20","slug":"a-store-stocks-8-different-kinds-of-batteris","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/a-store-stocks-8-different-kinds-of-batteris\/","title":{"rendered":"A store stocks 8 different kinds of batteris"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">A store stocks 8 different kinds of batteris, one of which is AA, with at least 28 packages of each kind how many ways can a purchase of 28 packages be made?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\"><strong>The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We are given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A store stocks <strong>8 different kinds of batteries<\/strong>, including <strong>AA<\/strong>.<\/li>\n\n\n\n<li>Each kind has <strong>at least 28 packages<\/strong>, so availability is not a limiting factor.<\/li>\n\n\n\n<li>A customer wants to buy <strong>exactly 28 packages<\/strong> in total.<\/li>\n\n\n\n<li>We are to find how many different <strong>combinations<\/strong> of 28 packages can be purchased, choosing from these 8 types.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udd22 Step-by-Step Breakdown<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Let the 8 types of batteries be represented by variables: x1+x2+x3+x4+x5+x6+x7+x8=28x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 = 28<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>xix_i = number of packages of the ii-th type of battery purchased.<\/li>\n\n\n\n<li>Each xi\u22650x_i \\geq 0 (non-negative integers).<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">This is a <strong>&#8220;stars and bars&#8221;<\/strong> combinatorics problem \u2014 the number of non-negative integer solutions to the equation: x1+x2+\u22ef+xk=nx_1 + x_2 + \\dots + x_k = n<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">is given by: (n+k\u22121k\u22121)\\binom{n + k &#8211; 1}{k &#8211; 1}<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Here, n=28n = 28 and k=8k = 8: (28+8\u221218\u22121)=(357)\\binom{28 + 8 &#8211; 1}{8 &#8211; 1} = \\binom{35}{7}<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 Final Answer:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">(357)=6,\u2009\u2063724,\u2009\u2063520\\boxed{\\binom{35}{7} = 6,\\!724,\\!520}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udcd8 Explanation (300 words)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">This problem involves choosing how to distribute 28 indistinguishable packages of batteries among 8 different kinds. This is a classic example of a <strong>combinations with repetition<\/strong> problem in combinatorics.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Each different distribution (like 4 AA, 3 AAA, 5 D, etc.) is a valid way to purchase the 28 packages. We\u2019re not concerned with the order of purchase or the package appearance, just how the total is split across the 8 kinds.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Because the store has <strong>at least 28 of each type<\/strong>, we don\u2019t have to worry about running out of a specific kind. That\u2019s important \u2014 it lets us assume <strong>unlimited supply<\/strong>, and apply the <strong>stars and bars<\/strong> method.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Think of the 28 packages as 28 stars (*). We want to divide these stars into 8 groups (battery types), using 7 bars (|) to separate them. For example:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>****|**||*****|*|*******||***\n<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">This represents a possible distribution of the 28 packages among the 8 types.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The number of ways to place these 7 dividers among the 35 total slots (28 stars + 7 bars) is: (357)\\binom{35}{7}<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This is calculated as: (357)=35!7!\u22c528!=6,\u2009\u2063724,\u2009\u2063520\\binom{35}{7} = \\frac{35!}{7! \\cdot 28!} = \\boxed{6,\\!724,\\!520}<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus, <strong>6,724,520<\/strong> different purchase combinations are possible.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A store stocks 8 different kinds of batteris, one of which is AA, with at least 28 packages of each kind how many ways can a purchase of 28 packages be made? The correct answer and explanation is: We are given: \ud83d\udd22 Step-by-Step Breakdown Let the 8 types of batteries be represented by variables: x1+x2+x3+x4+x5+x6+x7+x8=28x_1 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-16645","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/16645","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=16645"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/16645\/revisions"}],"predecessor-version":[{"id":16646,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/16645\/revisions\/16646"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=16645"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=16645"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=16645"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}