{"id":3446,"date":"2023-01-06T20:41:00","date_gmt":"2023-01-07T00:41:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=3446"},"modified":"2024-10-25T10:22:07","modified_gmt":"2024-10-25T14:22:07","slug":"combination-challenge-2022-12-27","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=3446","title":{"rendered":"Combination Challenge &#8211; 2023\/01\/06"},"content":{"rendered":"\n<p>Prove $$\\sum_{k=1}^n\\binom{n}{k}\\binom{n-1}{k-1}=\\binom{2n-1}{n}$$<\/p>\n\n\n\n<a href=\"javascript:toggle_visibility('combo-chall-2023-01-06');\">\ud83d\udd11<\/a>\n\n<div id=\"combo-chall-2023-01-06\" style=\"display:none\">\n\n\n\n<p><strong>Proof: <\/strong>Rewrite the left side as: $$\\sum_{k=1}^n\\binom{n}{k}\\binom{n-1}{n-k}$$<\/p>\n\n\n\n<p>The above can be interpreted as the number of ways to choose $n$ balls from $2n-1$ distinct balls, with balls divided into two groups, one group with $k$ distinct balls, the other group with $n-k$ distinct balls. <\/p>\n\n\n\n<p>Therefore the above sum can be re-rewritten as $\\boxed{\\binom{2n-1}{n}}$.<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Prove $$\\sum_{k=1}^n\\binom{n}{k}\\binom{n-1}{k-1}=\\binom{2n-1}{n}$$ \ud83d\udd11 Proof: Rewrite the left side as: $$\\sum_{k=1}^n\\binom{n}{k}\\binom{n-1}{n-k}$$ The above can be interpreted as the number of ways to choose $n$ balls from $2n-1$ distinct balls, with balls divided into two groups, one group with $k$ distinct balls, &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=3446\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[10,14],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/3446"}],"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=3446"}],"version-history":[{"count":15,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/3446\/revisions"}],"predecessor-version":[{"id":4589,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/3446\/revisions\/4589"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3446"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3446"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3446"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}