{"id":160675,"date":"2024-11-03T08:07:29","date_gmt":"2024-11-03T08:07:29","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=160675"},"modified":"2024-11-03T08:07:30","modified_gmt":"2024-11-03T08:07:30","slug":"a-pizza-parlor-offers-8-different-toppings","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/11\/03\/a-pizza-parlor-offers-8-different-toppings\/","title":{"rendered":"A pizza parlor offers 8 different toppings"},"content":{"rendered":"\n<p>A pizza parlor offers 8 different toppings.<br>a. How many 3-topping pizzas could they put on their menu? Assume double toppings are not allowed. b. How many total pizzas are possible, with between 1 and 8 toppings?<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To solve the problem, we need to utilize the concept of combinations, as the order in which the toppings are chosen does not matter.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">a. How many 3-topping pizzas could they put on their menu?<\/h3>\n\n\n\n<p>To find the number of ways to choose 3 toppings from a selection of 8, we use the combination formula:<\/p>\n\n\n\n<p>[<br>C(n, r) = \\frac{n!}{r!(n &#8211; r)!}<br>]<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( n ) is the total number of items (toppings),<\/li>\n\n\n\n<li>( r ) is the number of items to choose (toppings for the pizza).<\/li>\n<\/ul>\n\n\n\n<p>In this case, ( n = 8 ) and ( r = 3 ).<\/p>\n\n\n\n<p>[<br>C(8, 3) = \\frac{8!}{3!(8 &#8211; 3)!} = \\frac{8!}{3! \\cdot 5!}<br>]<\/p>\n\n\n\n<p>Calculating this gives:<\/p>\n\n\n\n<p>[<br>C(8, 3) = \\frac{8 \\times 7 \\times 6}{3 \\times 2 \\times 1} = \\frac{336}{6} = 56<br>]<\/p>\n\n\n\n<p>So, there are <strong>56 different 3-topping pizzas<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">b. How many total pizzas are possible, with between 1 and 8 toppings?<\/h3>\n\n\n\n<p>To find the total number of pizzas with between 1 and 8 toppings, we will sum the combinations for each case (1 topping, 2 toppings, \u2026, up to 8 toppings).<\/p>\n\n\n\n<p>Calculating each case:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>( C(8, 1) = 8 )<\/li>\n\n\n\n<li>( C(8, 2) = \\frac{8 \\times 7}{2 \\times 1} = 28 )<\/li>\n\n\n\n<li>( C(8, 3) = 56 ) (calculated above)<\/li>\n\n\n\n<li>( C(8, 4) = \\frac{8 \\times 7 \\times 6 \\times 5}{4 \\times 3 \\times 2 \\times 1} = 70 )<\/li>\n\n\n\n<li>( C(8, 5) = C(8, 3) = 56 ) (by symmetry)<\/li>\n\n\n\n<li>( C(8, 6) = C(8, 2) = 28 )<\/li>\n\n\n\n<li>( C(8, 7) = C(8, 1) = 8 )<\/li>\n\n\n\n<li>( C(8, 8) = 1 )<\/li>\n<\/ol>\n\n\n\n<p>Now, summing these values gives:<\/p>\n\n\n\n<p>[<br>8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 = 255<br>]<\/p>\n\n\n\n<p>Thus, the total number of different pizzas possible, with between 1 and 8 toppings, is <strong>255<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Summary<\/h3>\n\n\n\n<p>To summarize, the pizza parlor can create <strong>56 different 3-topping pizzas<\/strong>, and the total number of pizzas possible with any combination of toppings from 1 to 8 is <strong>255<\/strong>. The use of combinations allows us to calculate these possibilities efficiently, demonstrating how selection works in a combinatorial context.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A pizza parlor offers 8 different toppings.a. How many 3-topping pizzas could they put on their menu? Assume double toppings are not allowed. b. How many total pizzas are possible, with between 1 and 8 toppings? The Correct Answer and Explanation is : To solve the problem, we need to utilize the concept of combinations, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-160675","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/160675","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=160675"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/160675\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=160675"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=160675"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=160675"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}