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Pg1}{\bkmkend Pg1}\par\pard\ql \li3537\sb0\sl-414\slmult0 \par\pard\ql\li3537\sb184\sl-414\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf3\f4\fs36 PERMUTASI DAN KOMBINASI \par\pard\qj \li851\sb0\sl-480\slmult0 \par\pard\qj\li851\ri555\sb32\sl-480\slmult0\fi720 \up0 \expndtw0\charscalex105 \ul0\nosupersub\cf5\f6\fs36 Sebuah permutasi dari sebuah himpunan obyek-obyek \line \up0 \expndtw0\charscalex97 berbeda adalah penyusunan berurutan dari obyek-obyek tersebut. \par\pard\ql \li851\sb0\sl-414\slmult0 \par\pard\ql\li851\sb141\sl-414\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf3\f4\fs36 Contoh 1 \par\pard\ql \li851\sb66\sl-414\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf5\f6\fs36 Misalkan S = \{1, 2, 3\}. Susunan 3 1 2 adalah sebuah permutasi dari \par\pard\ql \li851\sb66\sl-414\slmult0 \up0 \expndtw0\charscalex97 S. Susunan 3 2 adalah sebuah permutasi-2 (\ul0\nosupersub\cf6\f7\fs36 2-permutation\ul0\nosupersub\cf5\f6\fs36 ) dari \par\pard\ql \li851\sb66\sl-414\slmult0 \up0 \expndtw0\charscalex97 S. \par\pard\qj \li851\sb0\sl-480\slmult0 \par\pard\qj\li851\ri559\sb32\sl-480\slmult0\fi720 \up0 \expndtw0\charscalex97 Banyak permutasi-r dari himpunan dengan n obyek berbeda \up0 \expndtw0\charscalex98 dinyatakan sebagai P(n,r) dimana \par\pard\ql \li2304\sb75\sl-414\slmult0 \up0 \expndtw0\charscalex98 P(n,r) = n . (n - 1) . (n - 2) . (n - 3) . ... . (n - r + 1). \par\pard\ql \li851\sb0\sl-414\slmult0 \par\pard\ql\li851\sb132\sl-414\slmult0 \up0 \expndtw0\charscalex97 Jika r = n , maka \par\pard\qj \li2251\ri1956\sb0\sl-500\slmult0 \up0 \expndtw0\charscalex98 P(n,n) = n . (n - 1) . (n - 2) . (n - 3) . ... . (n - n + 1). \up0 \expndtw0\charscalex99 = n . (n - 1) . (n - 2) . (n - 3) . ... . 1 \par\pard\ql \li2515\sb47\sl-414\slmult0 \up0 \expndtw0\charscalex97 = n ! \par\pard\ql \li851\sb66\sl-414\slmult0 \up0 \expndtw0\charscalex97 atau ditulis Pn = n ! \par\pard\ql \li851\sb0\sl-414\slmult0 \par\pard\ql\li851\sb152\sl-414\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf3\f4\fs36 Contoh 2\par\pard\sect\sectd\sbknone\cols3\colno1\colw3678\colsr160\colno2\colw2200\colsr40\colno3\colw5692\colsr160\ql \li2292\sb75\sl-414\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf5\f6\fs36 P(8,3) =\par\pard\ql \li3420\sb0\sl-414\slmult0 \par\pard\ql \li3420\sb55\sl-414\slmult0 \up0 \expndtw0\charscalex97 =\par\pard\ql \li851\sb0\sl-414\slmult0 \par\pard\ql \li851\sb345\sl-414\slmult0 \up0 \expndtw0\charscalex97 Rumus umum:\par\pard\column \ql \li20\sb75\sl-414\slmult0\tx1124 \up0 \expndtw0\charscalex97 8. \up0 \expndtw0\charscalex97 7. 6 \tab \up0 \expndtw0\charscalex90 = 336\par\pard\ql \li29\sb109\sl-425\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf7\f8\fs37 8 \up0 \expndtw0\charscalex97 7 \up0 \expndtw0\charscalex97 6 \up0 \expndtw0\charscalex97 5 \up0 \expndtw0\charscalex97 4 \up0 \expndtw0\charscalex97 3 \up0 \expndtw0\charscalex97 2 \up0 \expndtw0\charscalex97 1\par\pard\ql \li441\sb84\sl-425\slmult0 \up0 \expndtw0\charscalex97 5 \up0 \expndtw0\charscalex97 4 \up0 \expndtw0\charscalex97 3 \up0 \expndtw0\charscalex97 2 \up0 \expndtw0\charscalex97 1\par\pard\column \ql \li6068\sb0\sl-425\slmult0 \par\pard\ql \li909\sb141\sl-425\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf8\f9\fs37 8 \up0 \expndtw0\charscalex97 !\par\pard\ql \li20\sb1\sl-374\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf5\f6\fs36 =\par\pard\ql \li538\sb0\sl-333\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf8\f9\fs37 (8 \up0 \expndtw0\charscalex80 \u-4051? \up0 \expndtw0\charscalex97 3 \up0 \expndtw0\charscalex97 ) \up0 \expndtw0\charscalex97 !\par\pard\sect\sectd\sbknone\cols3\colno1\colw4355\colsr160\colno2\colw2713\colsr20\colno3\colw4522\colsr160\ql \li1687\sb333\sl-414\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf5\f6\fs36 n . (n-1) . (n-2) =\par\pard\column \ql \li20\sb92\sl-379\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf9\f10\fs33 n \up0 \expndtw0\charscalex97 .(n \up0 \expndtw0\charscalex97 -\up0 \expndtw0\charscalex96 1).(n \up0 \expndtw0\charscalex97 -\up0 \expndtw0\charscalex97 2). \up0 \expndtw0\charscalex94 (n\par\pard\ql \li1199\sb51\sl-379\slmult0 \up0 \expndtw0\charscalex96 (n \up0 \expndtw0\charscalex97 -\up0 \expndtw0\charscalex97 3).(n \up0 \expndtw0\charscalex95 -\par\pard\ql \li1767\sb116\sl-425\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf10\f11\fs37 n \up0 \expndtw0\charscalex97 !\par\pard\ql \li102\sb1\sl-365\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf5\f6\fs36 P(n,r) =\par\pard\column \ql \li20\sb92\sl-379\slmult0\tx2121 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf9\f10\fs33 -\up0 \expndtw0\charscalex97 3).(n \up0 \expndtw0\charscalex97 -\up0 \expndtw0\charscalex98 4)\tab \up0 \expndtw0\charscalex98 2. \up0 \expndtw0\charscalex97 1\par\pard\ql 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Pg2}{\bkmkend Pg2}\par\pard\qj \li851\sb0\sl-490\slmult0 \par\pard\qj\li851\ri560\sb46\sl-490\slmult0\fi720 \up0 \expndtw0\charscalex102 \ul0\nosupersub\cf5\f6\fs36 Sebuah kombinasi-r elemen-elemen dari sebuah himpunan \up0 \expndtw0\charscalex98 adalah pemilihan tak berurutan (tanpa memperhatikan urutan) r \up0 \expndtw0\charscalex99 elemen dari himpunan tersebut. \par\pard\ql \li851\sb0\sl-414\slmult0 \par\pard\ql\li851\sb119\sl-414\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf3\f4\fs36 Contoh 3 \par\pard\qj \li851\ri554\sb32\sl-480\slmult0 \up0 \expndtw0\charscalex100 \ul0\nosupersub\cf5\f6\fs36 Jika S = \{1, 2, 3, 4\}, susunan \{ 1, 3, 4 \} adalah sebuah kombinasi-3 \up0 \expndtw0\charscalex101 dari S. \par\pard\li851\sb0\sl-414\slmult0\par\pard\li851\sb139\sl-414\slmult0\fi720\tx5962 \up0 \expndtw0\charscalex103 Banyaknya kombinasi-r\tab \up0 \expndtw0\charscalex103 (\ul0\nosupersub\cf6\f7\fs36 r-combinations\ul0\nosupersub\cf5\f6\fs36 ) dari sebuah\par\pard\li851\sb73\sl-414\slmult0\fi0 \up0 \expndtw0\charscalex103 himpunan dengan n obyek berbeda dinyatakan sebagai C(n,r)\par\pard\sect\sectd\sbknone\cols3\colno1\colw3175\colsr160\colno2\colw904\colsr60\colno3\colw7471\colsr160\ql \li851\sb349\sl-414\slmult0 \up0 \expndtw0\charscalex94 atau \ul0\super\cf12\f13\fs43 Cr\par\pard\ql \li1936\sb0\sl-315\slmult0 \up0 \expndtw0\charscalex95 n\ul0\nosupersub\cf5\f6\fs36 atau\par\pard\ql \li851\sb0\sl-414\slmult0 \par\pard\ql \li851\sb296\sl-414\slmult0 \up0 \expndtw0\charscalex97 Rumus umum:\par\pard\column \ql \li20\sb119\sl-345\slmult0 \up0 \expndtw0\charscalex80 \ul0\nosupersub\cf13\f14\fs30 \u-3866? \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf14\f15\fs30 n \up0 \expndtw0\charscalex80 \ul0\nosupersub\cf13\f14\fs30 \u-3850?\par\pard\ql \li20\sb1\sl-308\slmult0\tx356 \up0 \expndtw0\charscalex80 \u-3865? \tab \up0 \expndtw0\charscalex80 \u-3849?\par\pard\ql \li20\sb0\sl-270\slmult0 \up0 \expndtw0\charscalex80 \u-3864? \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf14\f15\fs30 r \up0 \expndtw0\charscalex80 \ul0\nosupersub\cf13\f14\fs30 \u-3848?\par\pard\ql \li3749\sb0\sl-368\slmult0 \par\pard\ql \li689\sb91\sl-368\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf16\f17\fs32 r\par\pard\ql \li434\sb0\sl-288\slmult0 \up0 \expndtw0\charscalex97 C\par\pard\ql \li689\sb0\sl-288\slmult0 \up0 \expndtw0\charscalex95 n\par\pard\column \ql \li4289\sb0\sl-368\slmult0 \par\pard\ql \li4289\sb0\sl-368\slmult0 \par\pard\ql \li4289\sb0\sl-368\slmult0 \par\pard\ql \li4289\sb0\sl-368\slmult0 \par\pard\ql \li789\sb66\sl-368\slmult0 \up0 \expndtw0\charscalex97 n \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf15\f16\fs32 !\par\pard\ql \li20\sb0\sl-288\slmult0 \up0 \expndtw0\charscalex80 \u-4035?\par\pard\ql \li284\sb0\sl-288\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf16\f17\fs32 r \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf15\f16\fs32 ! \up0 \expndtw0\charscalex97 ( \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf16\f17\fs32 n \up0 \expndtw0\charscalex80 \ul0\nosupersub\cf15\f16\fs32 \u-4051? \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf16\f17\fs32 r \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf15\f16\fs32 ) \up0 \expndtw0\charscalex97 !\par\pard\sect\sectd\sbknone\cols3\colno1\colw3940\colsr160\colno2\colw840\colsr40\colno3\colw6790\colsr160\ql \li851\sb310\sl-414\slmult0 \up0 \expndtw0\charscalex94 \ul0\nosupersub\cf5\f6\fs36 Jika r = n, maka\par\pard\ql \li851\sb0\sl-414\slmult0 \par\pard\ql \li851\sb343\sl-414\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf3\f4\fs36 Contoh 4\par\pard\column \ql \li292\sb1\sl-336\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf17\f18\fs35 n\par\pard\ql \li20\sb0\sl-315\slmult0\tx563 \up0 \expndtw0\charscalex97 C \tab \up0 \expndtw0\charscalex80 \ul0\nosupersub\cf18\f19\fs35 \u-4035?\par\pard\ql \li292\sb0\sl-315\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf17\f18\fs35 n\par\pard\column \ql \li616\sb1\sl-387\slmult0 \up0 \expndtw0\charscalex97 n \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf18\f19\fs35 !\par\pard\ql \li1633\sb0\sl-315\slmult0 \up0 \expndtw0\charscalex80 \u-4035? \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf17\f18\fs35 1\par\pard\ql \li20\sb0\sl-315\slmult0 \up0 \expndtw0\charscalex97 n \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf18\f19\fs35 ! \up0 \expndtw0\charscalex97 ( \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf17\f18\fs35 n \up0 \expndtw0\charscalex80 \ul0\nosupersub\cf18\f19\fs35 \u-4051? \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf17\f18\fs35 n \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf18\f19\fs35 ) \up0 \expndtw0\charscalex97 ! \par\pard\sect\sectd\sbknone \ql \li851\sb83\sl-414\slmult0 \up0 \expndtw0\charscalex95 \ul0\nosupersub\cf5\f6\fs36 Misalkan S = \{1, 2, 3, 4\}. \par\pard\ql \li7346\sb55\sl-379\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf20\f21\fs33 4\par\pard\sect\sectd\sbknone\cols2\colno1\colw6927\colsr160\colno2\colw4673\colsr160\ql \li851\sb3\sl-330\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf5\f6\fs36 Kombinasi-4 dari S adalah \{ 1, 2, 3, 4 \},\par\pard\column \ql \li20\sb0\sl-297\slmult0\tx572 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf19\f20\fs33 C\tab \up0 \expndtw0\charscalex80 \ul0\nosupersub\cf20\f21\fs33 \u-4035? \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf19\f20\fs33 1\par\pard\ql \li279\sb0\sl-297\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf20\f21\fs33 4 \par\pard\sect\sectd\sbknone \ql \li851\sb27\sl-331\slmult0 \up0 \expndtw0\charscalex95 \ul0\nosupersub\cf5\f6\fs36 Kombinasi-3 dari S adalah \{ 1, 2, 3\}, \{1, 2, 4\}, \{2, 3, 4\}, \{1, 3, 4\}, \par\pard\ql \li1168\sb61\sl-414\slmult0 \up0 \expndtw0\charscalex97 \ul0\nosupersub\cf22\f23\fs36 3 \par\pard\li897\sb0\sl-324\slmult0\fi0\tx1497\tx1792 \up0 \expndtw0\charscalex86 \ul0\nosupersub\cf21\f22\fs36 C\tab \up0 \expndtw0\charscalex80 \ul0\nosupersub\cf22\f23\fs36 \u-4035?\tab \up0 \expndtw0\charscalex86 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