{"id":460,"date":"2012-03-15T19:34:05","date_gmt":"2012-03-16T00:34:05","guid":{"rendered":"http:\/\/linesoftangency.wordpress.com\/?p=460"},"modified":"2012-03-15T19:34:05","modified_gmt":"2012-03-16T00:34:05","slug":"0rganized-emptiness","status":"publish","type":"post","link":"http:\/\/blog.chrislusto.com\/?p=460","title":{"rendered":"0!rganized Emptiness"},"content":{"rendered":"

On the back of the fundamental counting principle<\/a>, my class has just established the fact that we can use n<\/em>!<\/strong> to count the number of possible arrangements of n<\/strong><\/em> unique objects.\u00a0 This is fantastic, but we don't always want to arrange all<\/em> of the n<\/em> things available to us, which is okay.\u00a0 We've also been introduced to the permutation function<\/a>, which has the very nice property of counting ordered arrangements of r<\/strong><\/em>-sized subsets of our n<\/em> objects.\u00a0 Handy indeed.<\/p>\n

Today we made an interesting observation: we now have not one, but two<\/strong> ways to count arrangements of, let's say, 7<\/strong> objects.<\/p>\n

    \n
  1. We can fall back on our old friend, the factorial<\/a>, and compute 7!<\/strong><\/li>\n
  2. We can use our new friend, the permutation function, and compute $latex bf{_7P_7}$<\/li>\n<\/ol>\n

    Since both expressions count the same thing, they ought to be equal, but then we run into this interesting tidbit when we evaluate (2):<\/p>\n

    $latex _7P_7 = frac{7!}{(7-7)!} = frac{7!}{0!}&s=2$,<\/p>\n

    which seems to imply that 0! = 1<\/strong>.\u00a0 To say this is counterintuitive for my kids would be a severe understatement.\u00a0 And in this moment of philosophical crisis, when the book might present itself as a palliative ally, students are instead met with this:<\/p>\n

    \"\"<\/a><\/p>\n

    To prevent inconsistency?\u00a0 How in the world are kids supposed to trust a mathematical resource that paints itself into a corner, only tacitly admits such, and then drops a bomb of a deus<\/em> ex<\/em> machina<\/em> in order to save face?\u00a0 I haven't been so angry since the ending of Lord of the Flies<\/em><\/a>.\u00a0 Especially when this problem appears two pages later:<\/p>\n

    \"\"<\/a><\/p>\n

    Okay, 8!<\/strong>.\u00a0 So how many ways can I arrange my bookshelf with a zero-volume<\/strong> reference set?\u00a0 One<\/strong>: I can arrange an empty shelf in exactly one way.\u00a0 And, since we already know that n<\/em>! counts the ways I can arrange n<\/em> objects, it follows naturally that this 1 way of arranging 0 things must also be represented by 0!<\/strong>.<\/p>\n

    There are a lot of good proofs\/justifications available for the willing Googler, but this one, to me, seems like the most natural and straightforward for a high school classroom.\u00a0 At a bare minimum, it's much, much better than, \"Because I need it to be true for my own convenience.\"<\/p>\n

    Only a math textbook could take something so lovely and make it seem dirty.<\/p>\n","protected":false},"excerpt":{"rendered":"

    On the back of the fundamental counting principle, my class has just established the fact that we can use n! to count the number of possible arrangements of n unique objects.\u00a0 This is fantastic, but we don't always want to arrange all of the n things available to us, which is okay.\u00a0 We've also been […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[32,43,55,56],"class_list":["post-460","post","type-post","status-publish","format-standard","hentry","category-math-teaching","tag-math","tag-permutations","tag-teaching","tag-textbooks"],"_links":{"self":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/460","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=460"}],"version-history":[{"count":0,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/460\/revisions"}],"wp:attachment":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=460"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=460"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=460"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}