{"id":1187,"date":"2017-11-20T00:02:29","date_gmt":"2017-11-20T00:02:29","guid":{"rendered":"https:\/\/kellyfish.me\/?p=1187"},"modified":"2019-01-08T05:24:39","modified_gmt":"2019-01-08T05:24:39","slug":"permutations","status":"publish","type":"post","link":"https:\/\/kellyfish.me\/index.php\/2017\/11\/20\/permutations\/","title":{"rendered":"Permutations"},"content":{"rendered":"

[latexpage]<\/span><\/p>\n

Fun Math for Girls
\n<\/span>By Kelly Tan<\/span><\/span><\/p>\n

\"\"<\/span><\/p>\n

Do you like making necklaces? It is a very interesting craft, and making necklaces with beads is easy. You can give these necklaces to friends as gifts or wear them yourself.<\/span><\/p>\n

It would be nice to make many different colored necklaces out of your collection of beads. Let’s count how many ways you can make a necklace from the beads that you have. Suppose you have 3 colors of beads: red(r), green(g), and blue(b). You want to make a necklace out of 3 different colored beads. This means that you can make the following patterns:<\/span><\/p>\n

rgb
\n<\/span>rbg
\n<\/span>grb
\n<\/span>gbr
\n<\/span>brg
\n<\/span>bgr<\/span><\/span><\/p>\n

Exactly 6 types. So, how did we exactly get these 6 patterns? The first bead could be red, green, or blue. So there are 3 possible types. For each of these colors, when you put the second bead in, you now have 2 possible colors to choose from, assuming you don’t want to reuse the same color. So, you have $3^.2$ possible patterns when you make a 2 beaded necklace. Finally, the last bead has to be chosen from the remaining color, which has only 1 choice. So you have a total of $3^.2^.1=6$ types.<\/span><\/p>\n

Remember the symbol for the number \u201c$3^.2^.1$\u201d is \u201c$3!$\u201d (yes, an exclamation point!). This value is called 3 factorial.<\/span><\/p>\n

It is very likely that you have many colors. Let’s say you have 20 colors of beads, and still want to make a 3 beaded necklace using different colors. How many types could you make? Well the same counting still applies, with some modification. For the first bead, you have 20 choices instead of 3 choices. Then for the second bead, you have 19 choices, and for the final bead, you have 18 choices. So the total number of possible types of necklace is $20^.19^.18=6840$. Wow! There are a lot of patterns you can make.<\/span><\/p>\n

So, how do you represent the number $20^.19^.18$? No, you can’t write it as $20!$, because 20 factorial is a very large number: $20^.19^.18^.17^.16….^.3^.2^.1$. The number is so large it will probably overflow your calculator. The symbol to represent that number is $_2_0P_3$<\/span>.<\/span><\/span><\/p>\n

So $_2_0P_3$\u00a0<\/span>$= 20^.19^.18$, but it can also be rewritten as:<\/span><\/span><\/p>\n

\\[\\frac{20^.19^.18^.(17^.16….^.3^.2^.1)}{(17^.16….^.3^.2^.1)}=\\frac{20!}{17!}\\] <\/span><\/p>\n

The number 17 is obtained from 20-3=17. In general, if you have 2 integers n and r, then\u00a0 \\[_nP_r\u00a0= \\frac{n!}{(n-r)!}\\]<\/span><\/p>\n

Consider another scenario. Suppose you don’t mind the beads in the necklace having the same color. What will be the number of patterns you can make if you have 20 different color beads?<\/span><\/p>\n

Well, for the first bead you put in, you still have 20 choices. For the second and third beads, you will still 20 choices each. So the total number of types you can make is $20^.20^.20=20^3$<\/span>. That is a huge number.<\/span><\/span><\/p>\n

To summarize: <\/span><\/p>\n

    \n
  1. The number of possible 3 beaded necklaces you can make from 3 possible colors, assuming there are no repeated colors, is: $3!$<\/span><\/li>\n
  2. The number of possible 3 beaded necklaces to make from 20 possible colors, assuming no repeated colors are allowed, is: $_2_0P_3$<\/span>\u00a0= $20^.19^.18=6840$.<\/span><\/span><\/li>\n
  3. The number of possible 3 beaded necklaces that you can make from 20 possible colors, assuming repeated colors are allowed, is: $20^.20^.20=20^3$<\/span>.<\/span><\/span><\/li>\n<\/ol>\n

    Finally, remember, this type of problem is called permutation, because the order of arrangement is important. A rgb necklace and a rbg necklace are considered different because the order of the beads are different. In the next article, we will talk about combination problems, where the order of arrangement is not important.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

    [latexpage] Fun Math for Girls By Kelly Tan Do you like making necklaces? It is a very interesting craft, and making necklaces with beads is easy. You can give these necklaces to friends as gifts or wear them yourself. It would be nice to make many different colored necklaces out of your collection of beads. […]<\/p>\n","protected":false},"author":1,"featured_media":1256,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[28],"tags":[],"class_list":["post-1187","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-articles"],"jetpack_featured_media_url":"https:\/\/i0.wp.com\/kellyfish.me\/wp-content\/uploads\/2017\/11\/s-l500.jpg?fit=500%2C400&ssl=1","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/posts\/1187","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/comments?post=1187"}],"version-history":[{"count":25,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/posts\/1187\/revisions"}],"predecessor-version":[{"id":1536,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/posts\/1187\/revisions\/1536"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/media\/1256"}],"wp:attachment":[{"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/media?parent=1187"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/categories?post=1187"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/tags?post=1187"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}