{"id":180,"date":"2019-12-31T19:53:36","date_gmt":"2019-12-31T19:53:36","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=180"},"modified":"2020-10-16T02:58:52","modified_gmt":"2020-10-16T02:58:52","slug":"mathcounts-exercises-8","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=180","title":{"rendered":"MATHCOUNTS Exercises – 8"},"content":{"rendered":"\n

3 cards are randomly selected from a desk of 9 cards uniquely number from 1 to 9. What is the probability that no two of the cards selected have numbers that differ by 1? Click here to show the solutions.<\/a><\/p>\n\n\n\n

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Solution 1 <\/strong>Let’s denote $f(m)$ as total number of 3-card groups that can be selected with $m$ as the smallest number in each of the groups, and $g(m,n)$ as the total number of 3-card groups that can be selected with $m$ as the smallest number and $n$ as the second smallest number. Each group meets the requirement that no two numbers differ by 1.<\/p>\n\n\n\n

Therefore, we have<\/p>\n\n\n\n

$$f(1)=\\sum_{i=3}^{7}g(1,i)=5+4+3+2+1=15$$ $$f(2)=\\sum_{i=4}^{7}g(2,i)=4+3+2+1=10$$ $$f(3)=\\sum_{i=5}^{7}g(3,i)=3+2+1=6$$ $$f(4)=\\sum_{i=6}^{7}g(4,i)=2+1=3$$ $$f(5)=\\sum_{i=7}^{7}g(5,i)=1$$ $$f(6)=f(7)=f(8)=f(9)=0$$<\/p>\n\n\n\n

Therefore, the total number of 3-card groups that meet the restriction is $$\\sum_{i=1}^{9}f(i)=15+10+6+3+1+0+0+0+0=35$$ Therefore the answer to the question is $$\\dfrac{35}{C(9,3)}=\\dfrac{35}{\\dfrac{9\\times 8\\times 7}{3\\times 2\\times 1}}=\\dfrac{5}{12}$$<\/p>\n\n\n\n

Solition 2<\/strong> Let’s consider 3-card groups that do not meet the requirement. There are two cases: one with only one pair of numbers differing by 1, such as (1, 2, 4); the other case with two pairs of numbers differing by 1, such as (2, 3, 4).<\/p>\n\n\n\n

For the first case, denote $h(x, y)$ is the number of 3-groups with only pair of numbers differing by $1$, $y = x + 1$ . Therefore the total number of 3-card groups with only one pair numbers differing by 1 is $$h(1,2)+h(2,3)+h(3,4)+h(4,5)+h(5,6)+h(6,7)+h(7,8)+h(8,9)$$ $$=6+5+5+5+5+5+5+6=42$$<\/p>\n\n\n\n

For the second case, there are 7 3-card groups: $(1,2,3), (2,3,4), \u2026, (7,8,9)$. Therefore, the total number of 3-card groups that do not meet the restriction is $42 + 7 = 49$.<\/p>\n\n\n\n

Therefore the answer to the question is $$1-\\dfrac{49}{C(9,3)}=1-\\dfrac{49}{\\dfrac{9\\times 8\\times 7}{3\\times 2\\times 1}}=\\dfrac{5}{12}$$<\/p>\n\n\n\n<\/div>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

3 cards are randomly selected from a desk of 9 cards uniquely number from 1 to 9. What is the probability that no two of the cards selected have numbers that differ by 1? Click here to show the solutions. … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[10,11,8],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/180"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=180"}],"version-history":[{"count":12,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/180\/revisions"}],"predecessor-version":[{"id":466,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/180\/revisions\/466"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=180"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=180"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=180"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}