If a 5-card hand is dealt from a standard 52 card deck, how many hands will contain exactly 2 aces? (not probability!)<\/p>\n\n\n\n
The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n To calculate the number of 5-card hands that contain exactly 2 aces, the problem can be broken down into two parts: choosing the 2 aces and selecting the remaining 3 cards from the non-ace cards.<\/p>\n\n\n\n A standard deck has 4 aces (one for each suit: hearts, diamonds, clubs, and spades). The number of ways to select 2 aces from these 4 is calculated using the combination formula: Combinations of 2 aces=(42)=4\u00d732\u00d71=6\\text{Combinations of 2 aces} = \\binom{4}{2} = \\frac{4 \\times 3}{2 \\times 1} = 6<\/p>\n\n\n\n So, there are 6 different ways to choose 2 aces from the 4 aces.<\/p>\n\n\n\n After choosing 2 aces, there are 50 cards left in the deck (since 4 aces were originally in the deck). Out of these 50 cards, 3 must be selected for the hand. These remaining cards cannot be aces, so we are choosing from the 48 non-ace cards. The number of ways to select 3 cards from these 48 non-ace cards is given by: Combinations of 3 non-ace cards=(483)=48\u00d747\u00d7463\u00d72\u00d71=17296\\text{Combinations of 3 non-ace cards} = \\binom{48}{3} = \\frac{48 \\times 47 \\times 46}{3 \\times 2 \\times 1} = 17296<\/p>\n\n\n\n So, there are 17,296 ways to select the remaining 3 non-ace cards.<\/p>\n\n\n\n To find the total number of hands containing exactly 2 aces, multiply the number of ways to choose the aces by the number of ways to choose the remaining cards: Total hands with 2 aces=(42)\u00d7(483)=6\u00d717296=103776\\text{Total hands with 2 aces} = \\binom{4}{2} \\times \\binom{48}{3} = 6 \\times 17296 = 103776<\/p>\n\n\n\n Therefore, there are 103,776<\/strong> possible 5-card hands that contain exactly 2 aces.<\/p>\n","protected":false},"excerpt":{"rendered":" If a 5-card hand is dealt from a standard 52 card deck, how many hands will contain exactly 2 aces? (not probability!) The correct answer and explanation is: To calculate the number of 5-card hands that contain exactly 2 aces, the problem can be broken down into two parts: choosing the 2 aces and selecting […]<\/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-241981","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/241981","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=241981"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/241981\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=241981"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=241981"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=241981"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Choosing the 2 aces<\/h3>\n\n\n\n
Step 2: Choosing the remaining 3 cards<\/h3>\n\n\n\n
Step 3: Total number of hands<\/h3>\n\n\n\n