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sol_ch7_countprinciples

sol_ch7_countprinciples - .kdswhu.rxqwlqj Sulqflsohv:14 41...

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Fkdswhu : Frxqwlqj Sulqflsohv :14 Edvlf Frxqwlqj Sulqflsoh 41 Wkhuh duh 7 urxwhv iurp Qhz \run wr Fklfdjr/ 8 urxwhv iurp Fklfdjr wr Ghqyhu/ dqg 6 urxwhv iurp Ghqyhu wr Orv Dqjohv1 Khqfh/ e| wkh pxowlsolfdwlrq sulqflsoh/ wkhuh duh 7 8 [email protected] gl huhqw urxwhv iurp Qhz \run wr Orv Dqjohv1 61 Wkhuh duh 7 w|shv ri vriw gulqnv/ 6 w|shv ri ghvhuwv/ dqg 8 w|shv ri sl}}dv1 D jxhvw uvw fdq fkrrvh d vriw gulqn/ wkhq d ghvhuw/ dqg wkhq d sl}}d1 E| wkh pxowlsolfdwlrq sulqflsoh/ wkhuh duh 7 6 [email protected] zd|v d glqqhu fdq eh vhuyhg1 81 Ohw + d> e , ghqrwhv dq rxwfrph vxfk wkdw wkh qxpehu d lv vkrzq e| wkh uvw glh dqg wkh qxpehu e lv vkrzq e| wkh vhfrqg glh1 Qrz wkh uhtxluhphqw lv wkdw d . e @; = Lw iroorzv wkdw wkh rxwfrphv duh +5 > 9, > +6 > 8, > +7 > 7, > +8 > 6, > dqg +9 > 5, = Khqfh/ wkhuh duh 8 rxwfrphv1 :1 Ohw + d> e , ghqrwhv dq rxwfrph vxfk wkdw wkh qxpehu d lv vkrzq e| wkh uvw glh dqg wkh qxpehu e lv vkrzq e| wkh vhfrqg glh1 Ohw D @ i + d> e , m d . e @6 > 4 d 9 > 4 e 9 j dqg E @ i + d> e , m d . e @9 > 4 d 9 > 4 e 9 j = Wkhq D @ i +4 > 5, > +5 > 4, j dqg E @ i +4 > 8, > +5 > 7, > +6 > 6, > +7 > 5, > +8 > 4, j = Fohduo|/ D _ E @ > = Wkxv/ wkh uhtxluhg qxpehu ri rxwfrphv lv m D m . m E m @[email protected]: = <1 D olfhqfh sodwh zlwk 6 ohwwhuv iroorzhg e| 7 gljlwv lv ri wkh irup o 4 o 5 o 6 g 4 g 5 g 6 g 7 > zkhuh o l uhsuhvhqwv d ohwwhu dqg g l uhsuhvhqwv d gljlw1 Ehfdxvh uhshwlwlrqv duh doorzhg/ wkhuh duh 59 fkrlfhv iru hdfk ri wkh ohwwhuv dqg 43 fkrlfhv iru hdfk ri wkh gljlwv1 Khqfh/ e| wkh pxowlsolfdwlrq sulqflsoh/ wkh qxpehu ri olfhqfh sodwhv lv 59 59 59 43 43 43 43 @ 59 6 43 7 @4:8 > :93 > 333 = 441 D olfhqfh sodwh zlwk 6 ohwwhuv iroorzhg e| 7 gljlwv lv ri wkh irup o 4 o 5 o 6 g 4 g 5 g 6 g 7 > zkhuh o l uhsuhvhqwv d ohwwhu dqg g l uhsuhvhqwv d gljlw1 Ehfdxvh rqh ri wkh ohwwhuv pxvw eh D> wkhuh duh 6 zd|v ri fkrrvlqj rqh ohwwhu wr eh D= Iru h{dpsoh/ o 4 fdq eh D> o 5 fdq eh D> ru o 6 fdq eh D= Ehfdxvh uhshwlwlrqv duh doorzhg/ hdfk ri wkh uhpdlqlqj 5 ohwwhuv fdq eh fkrvhq lq 59 zd|v dqg hdfk ri wkh 7 gljlwv fdq eh fkrvhq lq 43 zd|v1 Khqfh/ wkh qxpehu ri uhtxluhg olfhqfh sodwhv lv 6 59 59 43 43 43 43 @ 53 > 5;3 > 333 = Dqrwkhu zd| wr ghwhuplqh wkh qxpehu ri olfhqfh sodwhv lv dv iroorzv= Ohw [ eh wkh vhw ri olfhqfh sodwhv lq zklfk wkh uvw ohwwhu lv D> \ eh wkh vhw ri olfhqfh sodwhv lq zklfk wkh vhfrqg ohwwhu lv D> dqg ] eh wkh vhw ri olfhqfh sodwhv lq zklfk wkh wklug ohwwhu lv D= Ehfdxvh uhshwlwlrqv duh doorzhg/ lw iroorzv m [ m @59 5 43 7 @ m \ m @ m ] m = Wkxv/ wkh qxpehu ri olfhqfh sodwhv lv 6 59 5 43 7 = 461 Ohw v 4 v 5 v 6 v 7 v 8 v 9 v : v ; v < v 43 eh d vwulqj ri ohqjwk 43 = Qrz hdfk ri wkh elwv v l kdv 5 fkrlfhv1 Khqfh/ wkh qxpehu ri uhtxluhg vwulqjv li 5 43 @4357 = 481 Ohw v 4 v 5 v 6 v 7 v 8 v 9 v : v ; eh d vwulqj ri ohqjwk ; = D vwulqj ri ohqjwk ; wkdw pxvw hqg zlwk 43 pxvw eh ri wkh irup v 4 v 5 v 6 v 7 v 8 v 9 43 = Qrz hdfk ri v 4 >v 5 6 7 8 > dqg v 9 kdv 5 fkrlfhv1 Wkxv/ wkh qxpehu ri uhtxluhg vwulqjv lv 5 9 @97 = <4
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<5 :1 Frxqwlqj Sulqflsohv 4:1 Ohw v 4 v 5 v 6 v 7 v 8 v 9 eh d vwulqj ri ohqjwk 9 = D vwulqj ri ohqjwk 9 lq zklfk wkh vhfrqg elw lv 4 dqg wkh iwk elw lv 3 lv ri wkh irup v 4 4 v 6 v 7 3 v 9 = Qrz hdfk ri v 4 >v 6 7 > dqg v 9 kdv 5 fkrlfhv1 Khqfh/ wkh qxpehu ri uhtxluhg vwulqjv lv 5 7 @49 = 4<1 Ohw v 4 v 5 v 6 v 7 v 8 v 9 eh d vwulqj ri ohqjwk 9 =
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