{"id":222544,"date":"2025-05-31T12:48:17","date_gmt":"2025-05-31T12:48:17","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=222544"},"modified":"2025-05-31T12:48:19","modified_gmt":"2025-05-31T12:48:19","slug":"there-52-balls-in-a-box-16-red-16-blue-16-yellow-and-4-that-are-either-black-or-white","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/31\/there-52-balls-in-a-box-16-red-16-blue-16-yellow-and-4-that-are-either-black-or-white\/","title":{"rendered":"There 52 balls in a box: 16 red, 16, blue, 16 yellow, and 4 that are either black or white."},"content":{"rendered":"\n

There 52 balls in a box: 16 red, 16, blue, 16 yellow, and 4 that are either black or white. What is the least number of balls you must take out of the box (Without looking) to be sure that amount them there are 8 balls of the same color?
There 52 balls in a box: 16 red, 16, blue, 16 yellow, and 4 that are either black or white. What is the least number of balls you must take out of the box (Without looking) to be sure that amount them there are 8 balls of the same color?<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To be sure<\/strong> that you have 8 balls of the same color<\/strong>, the least number<\/strong> of balls you must take out of the box without looking<\/strong> is 37<\/strong>.<\/p>\n\n\n\n

\n\n\n\n

Explanation<\/strong><\/h3>\n\n\n\n

This is a classic pigeonhole principle<\/strong> problem in probability and combinatorics. We are given a total of 52 balls of five possible colors:<\/p>\n\n\n\n

    \n
  • 16 red<\/li>\n\n\n\n
  • 16 blue<\/li>\n\n\n\n
  • 16 yellow<\/li>\n\n\n\n
  • 4 black or white (unspecified distribution, but worst-case scenario must be considered)<\/li>\n<\/ul>\n\n\n\n

    Our goal is to determine the minimum number of balls<\/strong> we must blindly pick to guarantee<\/strong> that at least 8 balls<\/strong> are of the same color<\/strong>, no matter the actual outcome of our picks.<\/p>\n\n\n\n

    Step 1: Consider the worst-case scenario<\/h4>\n\n\n\n

    To delay reaching 8 of any single color, we can try to maximize variety<\/strong> in our picks.<\/p>\n\n\n\n

    Let\u2019s assume the worst-case scenario:<\/p>\n\n\n\n

      \n
    • You pick 7 red<\/strong>, 7 blue<\/strong>, 7 yellow<\/strong>, 4 black<\/strong>, and 4 white<\/strong> balls.<\/li>\n\n\n\n
    • That\u2019s a total of:
      7+7+7+4+4=297 + 7 + 7 + 4 + 4 = 29 balls, and still no 8 balls of any color<\/strong>.<\/li>\n<\/ul>\n\n\n\n

      But wait! There’s a problem: the total number of black and white balls is only 4<\/strong>, not 4 each. So, we can\u2019t have 4 black and<\/strong> 4 white in the worst case. The worst-case is if those 4 remaining balls are evenly split: 2 black and 2 white<\/strong>.<\/p>\n\n\n\n

      Step 2: Adjust for actual distribution<\/h4>\n\n\n\n

      So, the worst-case color distribution<\/strong> you could pull without getting 8 of one color is:<\/p>\n\n\n\n

        \n
      • 7 red<\/li>\n\n\n\n
      • 7 blue<\/li>\n\n\n\n
      • 7 yellow<\/li>\n\n\n\n
      • 2 black<\/li>\n\n\n\n
      • 2 white<\/li>\n<\/ul>\n\n\n\n

        That totals:
        7+7+7+2+2=257 + 7 + 7 + 2 + 2 = 25 balls<\/p>\n\n\n\n

        After 25 picks, it’s still possible<\/em> that you don’t have 8 of any color.<\/p>\n\n\n\n

        Step 3: Next pick ensures success<\/h4>\n\n\n\n

        On your 26th<\/strong> pick, no matter what color you draw<\/strong>, one of the first three main colors (red, blue, yellow) must reach 8, because you’ve already drawn 7 of each.<\/p>\n\n\n\n

        \u2705 Final Answer: 26 balls<\/strong> must be picked to guarantee<\/strong> 8 of the same color.<\/h3>\n\n\n\n
        \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

        There 52 balls in a box: 16 red, 16, blue, 16 yellow, and 4 that are either black or white. What is the least number of balls you must take out of the box (Without looking) to be sure that amount them there are 8 balls of the same color?There 52 balls in a box: […]<\/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-222544","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/222544","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=222544"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/222544\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=222544"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=222544"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=222544"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}