Combi66

COMBINATORIC and PROBABILITY PART 11 ʵ¤±~r݄ÎÌÔŶ…ʈÔÁzur~µÝ…ΚÁxÔËrÂÔ¢Ü~±cr   236 Bre ๏ лнр COMBINATORIC AND PROBABILITY>>>>>>>>>>>KRU BIRD

ª·›¸Á¦ ¸ ¥Š­´Áž¨¸É ¥œ ¨³ª·›¸‹´—®¤¼n 1. „‘Á„–”rÁ ºÊ °Š˜oœÁ„ ¸É ¥ª„´„µ¦œ´ Ĝ„µ¦šÎµŠµœÄ— Ç Ž¹ÉŠÂnŠÁž}œ k …´Êœ˜°œ ×¥…´Ê œš ¸É 1 Á¨º°„šÎµŠµœÅ—o n1ª·›¸ÄœÂ˜n¨³ª·›¸…°Š „µ¦šÎµŠµœÄœ…´Ê œ˜°œš ¸É 1 Á¨º°„šÎµŠµœÄœ…´Ê œ˜°œš ¸É ­°ŠÅ—o n2 ª·›¸Áž}œÁnœœ ¸ÊŞÁ¦ºÉ °¥ Ç ‹œ™¹Š… ´Ê œ˜°œ š¸É k Ž¹ÉŠ¤¸ª·›¸Á¨º°„šÎµŠµœÅ—o nkª·›¸ —´Šœ ´Ê œ­¦»žÅ—oªnµ ‹Îµœªœª·›¸šÎµŠµœ š´ÊŠ k …´Êœ˜°œ = n1 x n2 x n3 x … x nk ª·›¸ ˜³ª°¥nµŠ 1 Ĝ„µ¦Á¨º°„Å¡n 1 ĝ‹µ„­Îµ¦´ Ä®oŗo˜o¤ 4 ®¦º°Â˜o¤ 5 ¤¸ª·›¸Á¨º°„„¸Éª·›¸ ˜³ª°¥nµŠ 2 Ĝ„µ¦Á¨º°„Å¡n 2 ĝ‹µ„­Îµ¦´ Ä®oŗo˜o¤ 4 ¨³Â˜o¤ 5 ¤¸ª·›¸Á¨º°„„¸Éª·›¸ „µ¦®µ‹Îµœªœš¸É®µ¦‹ÎµœªœÁ˜È¤ª„¨Š˜´ª Ä® N o Ážœ‹} µœªœÁ˜ Î ¤ª„Ä— È Ç Á¦µ­µ¤µ¦™Á…¥œ¸ N Ä®°¥o Ĝ¦ ¼n žŸ¨‡ ¼ –…°Š‹ ¼ µœªœÁŒ¡µ³Å— Î o—´Šœ ¸Ê N = ... 3 3 2 2 1 1 CCC PPP … k k C .P Á¤ºÉ° P1 < P2 < P3 < … < Pk Áž}œ‹ÎµœªœÁŒ¡µ³ ¨³ C1 , C2 , C3, … , Ck Áž}œ‹ÎµœªœÁ˜È¤ª„ ­µ¤µ¦™­¦»žÅ—o—´Šœ ¸Ê 1. ¤¸‹Îµœªœœ´š ¸É ®µ¦ N ¨Š˜´ª Ášnµ„´ (C1 + 1) (C2 + 1) (C3 + 1) … (Ck + 1) ‹Îµœªœ 2. ¤¸‹ÎµœªœÁ˜È¤š ¸É ®µ¦ N ¨Š˜´ª Ášnµ„´ 2 (C1 + 1) (C2 + 1) (C3 + 1) … (Ck + 1) ‹Îµœªœ 3. ¤¸‹Îµœªœœ´š ¸ÉŤnčn‹ÎµœªœÁŒ¡µ³š ¸É ®µ¦ N ¨Š˜´ª Ášnµ„´ (C1 + 1) (C2 + 1) (C3 + 1) … (Ck + 1) ‹Îµœªœ ˜³ª°¥nµŠ 3 ¤¸‹Îµœªœœ´„¸É‹Îµœªœš ¸É ®µ¦ 100 ¨Š˜´ª ˜³ª°¥nµŠ 4 ¤¸‹Îµœªœœ´„¸É‹Îµœªœš ¸É ®µ¦ 210 ¨Š˜´ª ˜³ª°¥nµŠ 5 „ε®œ— N = 23 x 34 x 45 x 56 x 62 ¤¸‹Îµœªœœ´„¸É‹Îµœªœš ¸É ®µ¦ N ¨Š˜´ª ˜´ª°¥nµŠ 6 „ε®œ— M = 30,000,000 ‹Š®µªnµ 1. ¤¸‹Îµœªœœ´„¸É‹Îµœªœš ¸É ®µ¦ M ¨Š˜´ª 2. ¤¸‹ÎµœªœÁ˜È¤„¸É‹Îµœªœš ¸É ®µ¦ M ¨Š˜´ª 1. ¤¸‹Îµœªœœ´„¸É‹Îµœªœš ¸ÉŤnčn‹ÎµœªœÁŒ¡µ³š ¸É ®µ¦ M ¨Š˜´ª ←237 лн8 COMBINATORIC AND PROBABILITY>>>>>>>>>>>KRU BIRD

2 „µ¦®µ‹ÎµœªœÁ˜È¤ª„ k š¸É¤µ„š¸É­»— š¸ÉšÎµÄ®o ak ®µ¦ n! ¨Š˜´ª Á¤ºÉ° a Áž}œ‹ÎµœªœÁ˜È¤ª„ ¡·‹µ¦–µÁž}œ 2 „¦–¸ —´Šœ ¸Ê 1. a Áž}œ‹ÎµœªœÁŒ¡µ³ª„ 2. a Áž}œ‹ÎµœªœÁ˜È¤ª„Ž ¹É Š°¥¼nĜ¦¼žŸ¨‡¼–…°Š‹ÎµœªœÁŒ¡µ³ ˜³ª°¥nµŠ 1 ‹Š®µ‹ÎµœªœÁ˜È¤ª„ k š¸É¤µ„š¸É­»—š¸ÉšÎµÄ®o 2k ®µ¦ 20! ¨Š˜´ª ˜³ª°¥nµŠ 2 ‹Š®µ‹ÎµœªœÁ˜È¤ª„ k š¸É¤µ„š¸É­»—š¸ÉšÎµÄ®o 8k ®µ¦ 100! ¨Š˜´ª ˜³ª°¥nµŠ 3 ‹Š®µ‹ÎµœªœÁ˜È¤ª„ k š¸É¤µ„š¸É­»—š¸ÉšÎµÄ®o 13k ®µ¦ 2542! ¨Š˜´ª ˜³ª°¥nµŠ 4 ‹Š®µ‹ÎµœªœÁ˜È¤ª„ k š¸É¤µ„š¸É­»—š¸ÉšÎµÄ®o 6k ®µ¦ 40! ¨Š˜´ª ˜³ª°¥nµŠ 5 ‹Š®µ‹ÎµœªœÁ˜È¤ª„ k š¸É¤µ„š¸É­»—š¸ÉšÎµÄ®o 35k ®µ¦ 100! ¨Š˜´ª ˜³ª°¥nµŠ 6 ‹Îµœªœ«¼œ¥rš¸É¨Ššoµ¥ 1000! ¤µ„„ªnµ‹µœªœ« Î ¼œ¥rš¸É¨Ššoµ¥ 100! „¸É˜´ª ˜³ª°¥nµŠ 7 ‹Îµœªœœ´š ´Ê Š®¤—š ¸É ®µ¦ 12! ¨Š˜´ª ¤¸„¸É‹Îµœªœ ˜³ª°¥nµŠ 8 ‹Š®µ‹ÎµœªœÁ˜È¤ª„ k š¸É¤¸‡nµ¤µ„š¸É­»—š¸ÉšÎµÄ®o 288k ®µ¦ 100! ¨Š˜´ª¤¸‡nµÁšnµÄ— 2. ª·›¸Á¦¸¥Š­´Áž¨¸É ¥œ 2.1 ¢‡‡°Á¦¸¥¨ ´ Á¤ºÉ° n Áž}œ‹ÎµœªœÄ— Ç Â¨oª n! = 1 2 3 … n 2.2 ª·›¸Á¦¸¥Š­´ÁŽ˜Áž¨¸É¥œÁ·ŠÁ­oœ˜¦Š ´ ¤¸…°Š n ­·ÉŠš¸É˜nµŠ„´œœÎµ¤µÁ¦¸¥Š­´Áž¨¸É ¥œš´ÊŠ n ­·ÉŠ ª·›¸Á¦¸¥Š­´Áž¨¸É ¥œ = n! ª·›¸ ´ ¤¸…°Š n ­·ÉŠš¸É˜nµŠ„´œœÎµ¤µÁ¦¸¥Š­´Áž¨¸É ¥œ‡¦µª¨³ r ­·ÉŠ (r d n) ª·›¸Á¦¸¥Š­´Áž¨¸É ¥œ = Pn, r = )!( ! rn n  ª·›¸ ´ ¤¸…°Š n ­·ÉŠÃ—¥¤¸µŠ­ ·É ŠŽ ÎÊ µ„´œÁž}œ„¨»n¤ Ç k „¨»n¤ Ž¹ÉŠ„¨»n¤š ¸É 1 ŽÎʵ„´œ n1 ­·ÉŠ „¨»n¤š ¸É 2 ŽÎʵ„´œ n1 ­·ÉŠ … „¨»n¤ k ŽÎʵ„´œ nk ­·ÉŠ ¨³ n1 + n2 + n3 + … + nk = n ‹ÎµœªœÁ¦¸¥Š­´Áž¨¸É ¥œ…°Šš´ÊŠ n ­·ÉŠ = !!...! ! 21 k nnn n ª·›¸ ☆ 238 • лн9 COMBINATORIC AND PROBABILITY>>>>>>>>>>>KRU BIRD

3 2.3 ª·›¸Á¦¸¥Š­´ŠÁž¨¸É ¥œÁ·ŠªŠ„¨¤ ´ ¤¸…°Š n ­·ÉŠ ˜nµŠ„´œœÎµ¤µÁ¦¸¥Š­´Áž¨¸É ¥œÁ·ŠªŠ„¨¤ = (n - 1) ! ª·›¸ ´ ™oµ­ ·É Š…°Šš¸ÉœÎµ¤µÁ¦¸¥Š­´Áž¨¸É¥œÁž}œ­ ·É Š…°Šš ¸É ¡¨·„Å—o‹Îµœªœª·›¸Á¦¸¥Š­´Áž¨¸É ¥œ 2  )!1( n ´ ¤¸­·ÉŠ…°Š n ­·ÉŠœµ¤µÁ¦¸ Î ¥Š­´Áž¨¸É ¥œÂªŠ„¨¤‡¦µª¨³ r ­·ÉŠ ‹Îµœªœª·›¸Á¦¸¥Š­´Áž¨¸É ¥œ r ,rPn rrn n )!( !  2.4 ª·›¸Á¦¸¥Š­´Áž¨¸É¥œ…°Š®¨µ¥ž¦³Á£š­¨´„´œ Á·ŠÁ­oœ (­¨´ž¦³Á£š¨³ r ­·ÉŠ) ´ 2 ž¦³Á£šÁšnµ Ç „´œ ‹Îµœªœª·›¸ = 2 ! m ! m! ´ k ž¦³Á£šÁšnµ Ç „´œ ‹Îµœªœª·›¸ = k ! (m!)k Á·ŠªŠ„¨¤ (­¨´ž¦³Á£š¨³ 1 ­·ÉŠ) ´ 2 ž¦³Á£šÁšnµ Ç „´œ ‹Îµœªœª·›¸ = m ! (m - 1) ! ´ k ž¦³Á£šÁšnµ Ç „´œ ‹Îµœªœª·›¸ = km mk k)!(! 3. ª·›¸‹´—®¤¼n ¤¸…°Š n ­·ÉŠ˜nµŠ„´œ œÎµ¤µ‹´—®¤¼n‡¦µª¨³ r ­·ÉŠ ª·›¸‹´—®¤¼nš´ÊŠ®¤— = Cn, r !)!( ! rrn n  …o°­´ŠÁ„˜ 1. Cn, r = Cn, n - r !)!( ! rrn n  2. Cn, n = Cn, 0 = 1 3. Cn, 1 = Cn, n - 1 = n 4. „µ¦ÂnŠ­ ·É Š…°Šš ¸É Á®¤ º °œ„´œÂ¨oªÂ‹„ @ ™oµ¤¸…°Šš ¸É Á®¤º°œ„´œ n ­·ÉŠ ˜o°Š„µ¦Â‹„Ä®o‡œ r ‡œ ¤¸ª·›¸Â‹„—´Šœ ¸Ê 4.1 ‹„×¥š¸Éš»„‡œÅ—o¦´ ¤¸ª·›¸Â‹„ = Cn - 1, r - 1 )!1()!( )!1(   rrn n 4.2 ‹„×¥š¸ÉµŠ‡œÅ¤nŗo¦´ ¤¸ª·›¸Â‹„ = Cn + r - 1, r - 1 )!1(! )!1(   rn rn ๕ 239 • лой COMBINATORIC AND PROBABILITY>>>>>>>>>>>KRU BIRD

4 ‹µ„ª·›¸„µ¦Äœ…o°œ ¸Ê Á¦µ­µ¤µ¦™œÎµÅžž¦³¥»„˜rčo„´Á¦ºÉ°Ššª·œµ¤ Ĝ„µ¦š¸É ‹³®µ‹Îµœªœ¡‹œr ‹µ„„µ¦„¦³‹µ¥„Èŗo®¦º°­´¤ž¦³­·š› ·Í …°Š¡‹œr˜nµŠ Ç š¸É¤¸„ε¨´Š˜nµŠ Ç —´Šœ ¸Ê Ánœ ‹µ„„µ¦„¦³‹µ¥ (a + b + c)n ‹Îµœªœ¡‹œr‹µ„„µ¦„¦³‹µ¥ = Cn + 3 - 1, 3 - 1 !2! )!2( n n  „µ¦®µ ­.ž.­. …°Š 1 1 r x . 2 2 r x . 3 3 r x … kr x k ‹µ„„µ¦„¦³‹µ¥ (a1x1 + a2x2 + … akxk )n ­´¤ž¦³­·š› ·Í …°Š¡‹œrš¸É˜o°Š„µ¦ !!...! ! 21 k rrr n 1 1 r a . 2 2 r a … kr a k 5. „µ¦ÂnŠ„¨»n¤ @ ¤¸…°Š n ­·ÉŠš¸É˜nµŠ„´œ ¨³˜o°Š„µ¦ÂnŠ…°Šš´ÊŠ n ­·ÉŠ°°„Áž}œ„¨»n¤¥n°¥ Ç ‹Îµœªœ k „¨»n¤—´Šœ ¸Ê „¨»n¤š ¸É 1 ¤¸­·ÉŠ…°Š‹Îµœªœ n1 ­·ÉŠ „¨»n¤š ¸É 2 ¤¸­·ÉŠ…°Š‹Îµœªœ n2 ­·ÉŠ { „¨»n¤š ¸É k ¤¸­·ÉŠ…°Š‹Îµœªœ nk ­·ÉŠ ×¥š¸É n1 + n2 + n3 + … nk = n „µ¦®µ‹Îµœªœª·›¸„µ¦ÂnŠÂ¥„Áž}œ—´Šœ ¸Ê 5.1 ™oµ n1 , n2 , n3 , …, nk Ťn¤¸‹ÎµœªœÄ—Ášnµ„´œ ‹Îµœªœª·›¸„µ¦ÂnŠ !!...! ! 21 k nnn n ª·›¸ 5.2 ™oµ n1 , n2 , n3 , …, nk ¤¸µŠ‹Îµœªœš¸É¤¸‡nµÁšnµ„´œ ¨³„¨»n¤š¸É¤¸‹Îµœªœ­ ·É Š…°ŠÁšnµ„´œ ¤¸µŠ­ ·ÉŠÂ­—ŠÄ®oÁ®Èœ™¹Š‡ªµ¤Â˜„˜nµŠ¦³®ªnµŠ„¨n¤ ‹Îµœªœª·›¸„µ¦ÂnŠ !!...! ! 21 k nnn n ª·›¸ 5.3 ™oµ n1 , n2 , n3 , …, nk ¤¸µŠ‹Îµœªœš¸É¤¸‡nµÁšnµ„´œ ¨³„¨»n¤š¸É¤¸‹Îµœªœ­ ·É Š…°ŠÁšnµ„´œ Ťn¤¸µŠ­ ·ÉŠÂ­—ŠÄ®oÁ®Èœ™¹Š‡ªµ¤Â˜„˜nµŠ¦³®ªnµŠ„¨n¤ ‹Îµœªœª·›¸„µ¦ÂnŠ !!...!! ! 321 k nnnn n x яјзѬцеѠкѰђдшѠѯіѨѕјеѠклѼѥьњьдјҕєъѨѷєѨѝҕ ькеѠкѯъҕѥдѤ 1 ←240 • ло1 COMBINATORIC AND PROBABILITY>>>>>>>>>>>KRU BIRD

5 6. „µ¦Á¨ º °„…°Š ™oµ¤¸…°Š n ­·ÉŠš ¸É ˜„˜nµŠ„´œ ª·›¸„µ¦Á¨º°„…°Š°¥nµŠœo°¥ 1 ­·ÉŠ (Cn, 1 + Cn, 2 + Cn, 3 + … + Cn, n) = 2n - 1 ™oµ¤¸…°Š p + q + r + … ­·ÉŠ ×¥¤¸ p ­·ÉŠÁ®¤º°œ„´œ Áž}œœ·—š ¸É 1 q ­·ÉŠÁ®¤º°œ„´œ Áž}œœ·—š ¸É 2 r ­·ÉŠÁ®¤º°œ„´œ Áž}œœ·—š ¸É 3 ™oµ˜o°Š„µ¦Á¨º°„…°Š°¥nµŠœo°¥ œ·—¨³ 1 ­·ÉŠ = p x q x r … ª·›¸ ™oµ˜o°Š„µ¦Á¨º°„…°Š°¥nµŠœo°¥ 1 ­·ÉŠ = (p + 1) (q + 1) (r + 1) … - 1 ª·›¸ ™oµ¤¸…°Š p + q + r + … ­·ÉŠ ×¥¤¸ p ­·ÉŠÁ®¤º°œ„´œ Áž}œœ·—š ¸É 1 q ­·ÉŠÁ®¤º°œ„´œ Áž}œœ·—š ¸É 2 r ­·ÉŠÁ®¤º°œ„´œ Áž}œœ·—š ¸É 3 ™oµ˜o°Š„µ¦Á¨º°„…°Š°¥nµŠœo°¥ œ·—¨³ 1 ­·ÉŠ = (2p - 1) (2q - 1) (2r - 1) ª·›¸ ™oµ˜o°Š„µ¦Á¨º°„…°Š°¥nµŠœo°¥ 1 ­·ÉŠ = (2p . 2q . 2r . 1) = (2p+q+r - 1 ª·›¸ 7. š§¬‘¸šª·œµ¤ ™oµ n Áž}œ‹ÎµœªœÁ˜È¤ª„  n n n n n n ba n n ba n ba n ba n ba n ba 0 11 22 33 0 ... 0 1 2 3 ¸ ¹ · ¨ © § ¸ ¹ · ¨ © § ¸ ¹ · ¨ © § ¸ ¹ · ¨ © § ¸ ¹ · ¨ © §         ¦     ¸ ¹ · ¨ © § ¸ ¹ · ¨ © § ¸ ¹ · ¨ © § ¸ ¹ · ¨ © §     n k kkn n n n n n ba k n ba b n ba n ba n a 0 1 22 33 ... 1 2 3 ­·ÉŠš ¸É ‡ª¦¦¼o‹µ„„µ¦„¦³‹µ¥ (a + b)n 1. Ÿ¨…°Š„µ¦„¦³‹µ¥¤¸ n + 1 ¡‹œrÁ­¤° 2. Ÿ¨¦ª¤…°Š„ε¨´Š…°Š a ¨³ b Ĝ˜n¨³¡‹œr˜o°ŠÁšnµ„´ n 3. „ε¨´Š…°Š a Ĝ¡‹œr¦„‹³Á¦ ·É ¤‹µ„ n ¨oª¨—¨ŠÅžš¸¨³ 1 Ĝ˜n¨³¡‹œr™´—Åž‹œ„¦³š´É Š™¹Š 0 4. „ε¨´Š…°Š b Ĝ¡‹œr¦„‹³Á¦ ·É ¤‹µ„ 0 ¨oªÁ¡ ·É ¤… ¹Ê œš¸¨³ 1 Ĝ˜n¨³¡‹œr™´—Åž‹œ„¦³š´É Š™¹Š 0 5. ­´¤ž¦³­·š› ·Í …°Š¡‹œr¦„‹³Á¦ ·É ¤‹µ„ ¸ ¹ · ¨ © § 0 n ¨³¡‹œr™´—ÅžÁž}œ ¸ ¹ · ¨ © § 1 n , ¸ ¹ · ¨ © § 2 n ŞÁ¦ºÉ °¥ Ç ‹œ „¦³š ´É Š¡‹œr­»—šoµ¥‹³Áž}œ ¸ ¹ · ¨ © § n n ☆241 oa лол COMBINATORIC AND PROBABILITY>>>>>>>>>>>KRU BIRD

6 6. ™oµÂšœ a = 1 ¨³ b = 1 ŞĜ­¤„µ¦Á¦µ‹³Å—o¨´„¬–³š¸É­Îµ‡´‡º° ¸ ¹ · ¨ © § ¸ ¹ · ¨ © § ¸ ¹ · ¨ © § ¸ ¹ · ¨ © §  n nn n a n n ... 321 2 7. ¡‹œrš¸É r = Tr = )1( 1 1   ¸ ¹ · ¨ © §  rn r a b r n ˜nÁ¡ ºÉ °‡ªµ¤­³—ª„Á¦µœ·¥¤Äo¡‹œrš¸É r + 1 = Tr+1 = rrn ba r n  ¸ ¹ · ¨ © § 8. ™oµ n Áž}œ‹ÎµœªœÁ˜È¤ª„¨³ c < k < n ¨oª Cn,k = Cn-1,k-1 + Cn-1,k 9. Ck,k = Ck+1,k + Ck+2,k + … + Cn,k = Cn+1,k+1 ‡ªµ¤œ nµ‹³Áž}œ 1.1 „µ¦š—¨°Š­»n¤ Ž¤Ážd¨­ÁžŽ ¨³Á®˜»„µ¦– r „µ¦š—¨°Š­»n¤ (randome expriment) ®¤µ¥™¹Š „µ¦š—¨°ŠÄ— Ç š¸É¥´ŠÅ¤n­µ¤µ¦™šÎµœµ¥®¦º° ¦¼oŸ¨¨´¡›rš¸ÉÁ„·—…¹Êœ ˜n­µ¤µ¦™°„Å—oªnµŸ¨¨´¡›rš´ÊŠ®¤—š ¸É ‹³Á„·—… ¹Ê œ¤¸°³Å¦oµŠ Ž¤Ážd¨­ÁžŽ (sample space) ®¤µ¥™¹Š ÁŽ˜…°ŠŸ¨¨´¡›rš´ÊŠ®¤—Ášnµš ¸É ‹³Á„·—… ¹ÊœÅ—o‹µ„„µ¦ š—¨°Š­»nŠ Á®˜»„µ¦–r (event) ®¤µ¥™¹Š ­´ÁŽ˜Ä— Ç …°ŠÂŽ¤Ážd¨­ÁžŽ Á®˜»„µ¦–rš¸Éž¦³„°—oª¥Ÿ¨ ¨´¡›rÁ¡¸¥ŠŸ¨¨´¡›rÁ—¸¥ª Á¦µÁ¦¸¥„ªnµ Á®˜»„µ¦–r°¥nµŠŠnµ¥ (simple event) 1.2 ‡ªµ¤œnµ‹³Áž}œ…°ŠÁ®˜»„µ¦– r (probability of an event) Ž¤Ážd¨­ÁžŽš¸É ‹³„¨nµª˜n°Åžœ¸Ê™oµÅ¤nŗo¦³»Áž}œ°¥nµŠ°ºÉœ …o°˜„¨ŠªnµÂŽ¤Ážd¨­ÁžŽ ‹³Áž}œ ÁŽ˜‹Îµ„´—Ášnµœ´Êœ „ε®œ—Ä®o S = Ž¤Ážd¨­ÁžŽŽ¹É ŠÂ˜n¨³Ÿ¨¨´¡›rĜ S ¤¸Ã°„µ­Á„·—… ¹Ê œÁšnµ Ç „´œ E = Á®˜»„µ¦–rĜ S n(S) = ‹ÎµœªœŸ¨¨´¡›rĜ S n(E) = ‹ÎµœªœŸ¨¨´¡›rĜ E 242 ไร ☆ лом COMBINATORIC AND PROBABILITY>>>>>>>>>>>KRU BIRD

7 ‡ªµ¤œnµ‹³Áž}œ…°ŠÁ®˜»„µ¦–r E ˜n°Åž‹³Á…¸¥œÂšœ—oª¥­´¨´„¬–r P(E) Ž¹ÉŠ¤¸‡ªµ¤®¤µ¥—´Šœ ¸Ê       Sn En EP 1.3 ‡»–­¤´˜·…°Š‡ªµ¤œnµ‹³Áž}œ Ä®o S šœÂŽ¤Ážd¨­ÁžŽ ¨³ A šœÁ®˜»„µ¦–r‹³Å—oªnµ (1) 0 d P(A) d 1 (3) P(A) = 1 „Șn°Á¤ºÉ° A = S (2) P(A) = 0 „Șn°Á¤ºÉ° A = I (4) ™oµ A  B ¨oª P(A) d P(B) 1.4 ‡ªµ¤œnµ‹³Áž}œ…°Š¥¼Áœ¸¥œ…°ŠÁ®˜»„µ¦– r Ä®o S šœÂŽ¤Ážd¨­ÁžŽ ¨³ A, B šœÁ®˜»„µ¦–r ? A ‰ B ¨³ B  S ? (A ‰ B)  S —´Šœ ´Ê œÁ¦µ™º°ªnµ A ‰ B Áž}œÁ®˜»„µ¦–r®œ¹ÉŠ‹µ„‡ªµ¤¦¼oÁ„ ¸É ¥ª„´Á¦ ºÉ °ŠÁŽ˜ ‹³Å—oªnµ n(A ‰ B) = n(A) + n(B) - n(A ˆ B) )( )( )( )( )( )( )( )( Sn BAn Sn Bn Sn An Sn BAn ˆ  ‰ œ´Éœ‡º° P(A ‰ B) = P(A) + P(B) - P(A ˆ B) Ĝ„¦–¸ A ¨³ B Áž}œÁ®˜»„µ¦–rš¸ÉŤn¤¸Ÿ¨¨´¡›r¦nª¤„´œ ‹³Å—oªnµ A ˆ B = I P(A ˆ B) = 0 —´Šœ ´Ê œ‹µ„­¼˜¦ (1.4.1) ‹³Å—oªnµ P(A ‰ B) = P(A) + P(B) ←243 • лон COMBINATORIC AND PROBABILITY>>>>>>>>>>>KRU BIRD

8 1.5 ‡ªµ¤œnµ‹³Áž}œ…°Š‡°¤¡¨¸Á¤œ˜ r ¨³Ÿ¨˜nµŠ…°ŠÁŽ˜ u Ä®o S = Ž¤Ážd¨­ÁžŽ A Ä®o A Áž}œÁ®˜»„µ¦–rĜ S Ac ‹³Å—oªnµ ‰ c SAA ¨³ ˆ AA c I —´Šœ´Êœ ‰ c  APAPAAPSP c)()()()(1 œ´Éœ‡º° c  APAP )(1)( ressrn 244 • лоо COMBINATORIC AND PROBABILITY>>>>>>>>>>>KRU BIRD

COMBINATORIC and PROBABILITY PART 12 }rx˜·ÎÊΩ~µÝ…ΚÁxÔËrÂÔ¢Ü~±cr   Dtste 245 ๏ лоп COMBINATORIC AND PROBABILITY>>>>>>>>>>>KRU BIRD

…o°­°Á°œš¦µœŽ r ¨³Âœª…o°­° 1. ¤¸‹Îµœªœœ´„¸É‹Îµœªœš ¸É ®µ¦ 2544 ¨Š˜´ª 1. 6 ‹Îµœªœ 2. 10 ‹Îµœªœ 3. 20 ‹Îµœªœ 4. 40 ‹Îµœªœ 2. ‹Š®µŸ¨ª„…°ŠÁ¨… 4 ®¨´„š ¸É Á„·—‹µ„˜´ªÁ¨… 2, 3, 5, 7 ×¥š¸É ˜n¨³®¨´„ŤnŽÎʵ„´œ¤¸‡nµÁšnµÄ— 1. 113322 2. 166101 3. 221133 4. 332211 3. ‹Š®µ‹Îµœªœª·›¸š¸É‹³‹´Â¤ª 4 ˜´ª ¨³Á­º° 1 ˜´ªÄ­n„¦Š Ž¹ÉŠ¤¸„¦Š°¥¼nš´ÊŠ®¤— 4 „¦Š ×¥š¸É Á­º°‹³˜o°Š °¥¼n„¦ŠÄ—„¦Š®œ¹É ŠÁ¡¸¥Š˜´ªÁ—¸¥ªÁšnµœ ´Ê œ‹³¤¸„¸Éª·›¸ 1. 24 ª·›¸ 2. 48 ª·›¸ 3. 324 ª·›¸ 4. 423 ª·›¸ 4. ˜q°ŠÂ¨³­˜¸¢ …nŠ…´œ­œ»„Á„°¦r·Š¦µŠª´¨‡¦ ´Ê Š®œ¹ÉŠ ×¥¤¸„˜·„µªnµ™oµÄ‡¦œ³ 2 Á„¤­r˜·—˜n°„´œ Ÿ¼oœ´Êœ Áž}œŸ¼oœ³ ®¦º°Ä‡¦œ³ 3 Á„¤­r„n°œ Ÿ¼oœ´Êœ‹³Áž}œŸ¼oœ³‹Š®µ‹Îµœªœ¦¼žÂš´Ê Š®¤—…°Š„µ¦ …nŠ…´œ‹œ¤¸Ÿ¼oœ³ 1. 8 ª·›¸ 2. 9 ª·›¸ 3. 10 ª·›¸ 4. 11 ª·›¸ 5. ‹ŠÁ…¸¥œŸ¨‡¼–…°Š 1 x 3 x 5 x 7 x 9 x 11 x 13 x 15 Ä®o°¥¼nĜ¦¼žÂ¢„š°Á¦¸¥¨ 1. !6.2 !15 6 2. !8.2 !15 8 3. !6.2 !16 6 4. !8.2 !16 8 6. Á¨… 4 ®¨´„ š¸ÉÁ„·—‹µ„˜´ªÁ¨… 4 ˜´ª š¸É¤¸‡nµ¤µ„„ªnµ 5000 ×¥­¦oµŠ‹µ„Á¨…×— 1, 2, 3, 4, 5 ¨³ 6 ‹³¤¸ š´ÊŠ®¤—„¸É‹Îµœªœ ™oµ¤¸Á¨… 4 Ášnµœ´Êœ š¸ÉčoŽÎʵŗoœ°„œ ´Ê œ˜´ªÁ¨…Ĝ˜n¨³®¨´„ŤnŽÎʵ„´œ 1. 120 2. 144 3. 146 4. 148 7. ‹´—œ´„Á¦¸¥œ 6 ‡œ Ž¹ÉŠ¤¸Á¤˜˜µÂ¨³ž¦µ–¸¦ª¤°¥¼n—oª¥Ä®oÁ¦¸¥ŠÂ™ªÁž}œ 2  š ¸É ®œ¹ÉŠ œ´„Á¦¸¥œ š´ÊŠ®¤—¥ºœÁ¦¸¥ŠÁž}œÂ™ª˜¦Š ×¥š¸ÉÁ¤˜˜µÂ¨³ž¦µ–¸¥ºœ˜·—„´œ š ¸É ­°Šœ´„Á¦¸¥œš ´Ê Š®¤—¥ºœÁž}œ ªŠ„¨¤ ×¥š¸ÉÁ¤˜˜µÂ¨³ž¦µ–¸¥ºœ˜¦Š„´œ…oµ¤ ¨oª‹Îµœªœª·›¸…°Š„µ¦‹´—˜n¨³ÂÂ˜„˜nµŠ„´œ Ášnµ„´…o°Ä—˜n°Åžœ¸Ê 1. 96 2. 120 3. 196 4. 216 8. Ĝ„µ¦ž¦³»¤‡¦ ´Ê Š®œ¹ÉŠ ¤¸Ÿ¼ošœ‹µ„ 3 ž¦³Áš«Á…oµ¦nª¤ž¦³»¤ ×¥¤¸Ÿ¼ošœž¦³Áš«¨³ 3 ‡œ ‹Îµœªœª·›¸š´ÊŠ®¤—š ¸É ‹³‹´—Ä®oŸ¼ošœÂ˜n¨³ž¦³Áš«˜o°Šœ ´É Š˜n°„´œÄœ„µ¦ž¦³»¤Ã˜p³„¨¤Ášnµ„´ÁšnµÄ— 1. 144 2. 324 3. 432 4. 480 9. Ä®o A = {1, 2, 3} ¨³ B = {a, b, c, d} ¨oª‹Îµœªœ­¤µ·„…°ŠÁŽ˜ {f : A ¤ B ~ f ŤnÁž}œ¢{Š„r´œ 1-1} Ášnµ„´…o°Ä—˜n°Åžœ¸Ê 1. 40 2. 34 3. 30 4. 24 10.™oµ A = {1, 2, 3, 4, 5, 6} ¨³ B = {1, 2, 3} ¨oª‹Îµœªœ¢{Š„r´œ f : A ¤ B š´ÊŠ®¤—Ž¹ÉŠ f(1) z 1 ®¦º° f(2) z 2 ®¦º° f(3) z 3 Ášnµ„´…o°Ä—˜n°Åžœ¸Ê 1. 530 2. 612 3. 702 4. 814 ←246 лор COMBINATORIC AND PROBABILITY>>>>>>>>>>>KRU BIRD

10 11. „ε®œ—Ä®o A = {a1, a2, a3, a4} ¨³ B = {b1, b2, b3} ÁŽ˜ {f ~ f : C ¤ B š´ÊŠ®¤—Ž¹ÉŠ C  B ×¥ C z I} ¤¸‹Îµœªœ­¤µ·„Ášnµ„´…o°Ä— 1. 12 2. 81 3. 255 4. 405 12. „ε®œ—Á­oœ˜¦ŠÄœÂœªœ°œ…œµœ„´œ 5 Á­oœ œª— ·É Š…œµœ„´œ 7 Á­oœ ×¥¦³¥³®nµŠ¦³®ªnµŠÁ­oœ …œµœ…°Šš»„Âœª®nµŠÁšnµ„´œ ‡º° —oµœ¨³ 1 ®œnª¥ ‹Š®µ‹Îµœªœ¦¼ž­¸É Á®¨ ¸É ¥¤š ´Ê Š®¤—š ¸ÉŤnčn­¸ÉÁ®¨ ¸É ¥¤ ‹˜»¦´­ 1. 210 2. 160 3. 196 4. 178 13. Á­oœ˜¦Š»—®œ¹ÉŠ ž¦³„°—oª¥Á­oœ…œµœÂœªœ°œ 11 Á­oœ ×¥Á­oœš ¸É °¥¼nÁ¦¸¥Š„´œ®nµŠ„´œ 1 œ·Êª ¨³ Á­oœ˜¦Š°¸„»—®œ ¹ÉŠž¦³„°—oª¥Á­oœ…œµœÂœª˜´ÊŠ 10 Á­oœ ×¥Á­oœš ¸É °¥¼nÁ¦¸¥Š„´œ®nµŠ„´œ 1 œ·Êª ‹Îµœªœ¦¼ž­¸É Á®¨ ¸É ¥¤‹˜»¦´­š¸É¤¸¡ºÊœš ¸ÉŤnÁ„·œ 9 ˜µ¦µŠœ·Êª š´ÊŠ®¤—š ¸É Á„·—‹µ„„µ¦˜´—…°ŠÁ­oœ˜¦Š š´ÊŠ 2 »— œ¸ÊÁšnµ„´…o°Ä—˜n°Åžœ¸Ê 1. 110 2. 218 3. 308 4. 450 14. ‹´—¡œ´„Šµœ 6 ‡œ Áž}œ 3 „¨»n¤ nŠÅžšÎµŠµœ 3 Šµœ Ž¹ÉŠÂ˜„˜nµŠ„´œ ×¥‹´—„¨»n¤¨³„ ¸É ‡œ„Èŗo ‹Îµœªœª·›¸‹³‹´—Å—oÁšnµ„´…o°Ä—˜n°Åžœ¸Ê 1. 270 2. 540 3. 810 4. 1080 15. ‹´—Á—È„ 1 ‡œ Ÿ¼o®·Š 3 ‡œ ¨³Ÿ¼oµ¥ 3 ‡œ œ´ÉŠ¦°Ã˜p³„¨¤ ×¥š¸ÉŸ¼oµ¥Å¤nœ´ÉŠ˜·—„´Á—È„‹³‹´—Å—o š´ÊŠ®¤—„¸Éª·›¸ 1. 35 ª·›¸ 2. 48 ª·›¸ 3. 96 ª·›¸ 4. 144 Š·›¸ 16. „ε®œ—‹»— 7 ‹»—œÁ­oœ¦°ªŠ„¨¤ ‹³­¦oµŠ¦¼žÁ®¨¸É¥¤Å—oš´ÊŠ®¤—„¸É¦¼žÁ¤ºÉ°Ä®o‹»—œÁ­oœ¦°ªŠ Á®¨nµœ ¸ÊÁž}œ‹»—¥°— 1. 35 2. 42 3. 56 4. 99 17. µ¥ 3 ‡œ ¨³®·Š 3 ‡œ Á…oµ‡·ªÄœÂ™ªÁ—¸¥ª„´œÁ¡ ºÉ °Ž ºÊ °˜ ´Ì ª£µ¡¥œ˜¦r‡ªµ¤œnµ‹³Áž}œš ¸É ®·Šš´ÊŠ 3 ‡œ ‹³¥ºœÁ¦¸¥Š˜·—„´œš ´ÊŠ®¤—Ĝ™ª ¤¸‡nµÁšnµ„´ÁšnµÄ— 1. 0.1 2. 0.2 3. 0.3 4. 0.4 18. …o°­°»—®œ¹ÉŠ ¤¸ 2 ˜°œ ˜°œ¨³ 4 …o° ¤¸‡Îµ­ ´ÉŠÄ®oŸ¼ošÎµ…o°­°šÎµ…o°­°˜°œš ¸É 1 °¥nµŠœo°¥ 1 …o° ¨³šÎµ…o°­°˜°œš ¸É ­°Š 2 …o° ‹Îµœªœª·›¸š¸ÉŸ»o­°‹³šÎµ…o°­°»—œ¸Ê¤¸„¸Éª·›¸ 1. 16 2. 24 3. 36 4. 90 ☆247 • ло8 COMBINATORIC AND PROBABILITY>>>>>>>>>>>KRU BIRD

11 19. Ĝ„µ¦­¦oµŠÁ¤˜¦·„Žr »¼ º «¬ ª d b c a ×¥š¸É a, b, c  {-2, -1, 0, 1, 2} ¨³ d = 0 ‡ªµ¤œnµ‹³Áž}œš ¸Éŗo Á¤˜¦·„Žrœ°œ-Ž·Š„¼¨µ¦rÁž}œÁšnµÄ— 1. 0.25 2. 0.34 3. 0.52 4. 0.64 20. Ĝ„µ¦Á¨nœÁ„¤­ršµ¥˜´ªÁ¨…‹ÎµœªœÁ˜È¤ 4 ®¨´„Ťn¤¸„µ¦Ž ÎÊ µ ×¥˜´ªÁ¨…˜n¨³®¨´„ŤnÁž}œÁ¨…«¼œ¥r ‡ªµ¤œnµ‹³Áž}œš ¸É ‹³šµ¥˜´ªÁ¨…Å—o™¼„˜o°Š 4 ˜´ª ˜n®¨´„…°Š˜´ªÁ¨…Ÿ·—ÅžÁ¡¸¥Š 2 ®¨´„Ášnµœ´Êœ ¤¸‡nµ Ášnµ„´ÁšnµÄ— 1. 126 1 2. 252 1 3. 504 1 4. 504 3 21. ™oµ f Áž}œ¢{Š„r´œ‹µ„ÁŽ˜ x Ş¥´ŠÁŽ˜ y ×¥š¸É x = {10, 20, 30, 40} ¨³ y = {2, 4, 6, 8, 10, 12} ‡ªµ¤œnµ‹³Áž}œš ¸É f Áž}œ¢{Š„r´œ®œ ¹É Š˜n°®œ¸ÉŠ‹µ„ÁŽ˜ x Ş¥´ŠÁŽ˜ y ˜¦Š„´…o°Ä— 1. 18 5 2. 3 1 3. 18 7 4. 9 4 22. ™»ŠÄ®œ¹É Š¤¸¨¼„„oª…œµ—Á—¸¥ª„´œ°¥¼n 10 ¨¼„ Áž}œ­¸Â—Š 3 ¨¼„ ­¸…µª 5 ¨¼„ ­¸—ε 2 ¨¼„ ­»n¤®¥· ¨¼„„oª‹µ„™»Š 2 ‡¦´ÊŠ Ç ¨³¨¼„ ץŤnÄ­n‡ºœ ‡ªµ¤œnµ‹³Áž}œš ¸É ‹³®¥·Å—o¨¼„š ¸É 2 Áž}œ­¸Â—ŠÁšnµ„´ …o°Ä— 1. 3 1 2. 10 3 3. 100 27 4. 100 33 23. Ĝ„µ¦š°—¨¼„Á˜qµ 2 ¨¼„ ¡¦o°¤„´œ 1 ‡¦´ÊŠ ‡ªµ¤œnµ‹³Ážjœš ¸É Ÿ¨ª„…°ŠÂ˜o¤œ®œoµ…°Š¨¼„Á˜qµš´ÊŠ 2 ¨¼„‹³Áž}œÁ¨…š¸É®µ¦—oª¥ 4 Ťn¨Š˜´ª ¤¸‡nµÁšnµ„´…o°Ä— 1. 0.25 2. 0.50 3. 0.75 4. 0.95 24. „¨n°ŠÄ®œ¹É Š¤¸¨¼„®·œ­¸…µª 5 ¨¼„ ­¸Á…¸¥ª 3 ¨¼„ ­¸œÎʵÁŠ·œ 2 ¨¼„ ™oµ®¥·¨¼„®·œ°¥nµŠ­»n¤‡¦ ´Ê Š¨³ 1 ¨¼„ ץŤnÄ­n‡ºœ 3 ‡¦´ÊŠ ¨oª‡ªµ¤œnµ‹³Áž}œš ¸É ‹³®¥·Å—o¨¼„®·œ­¸Á—¸¥ª„´œ°¥nµŠœo°¥ 2 ¨¼„ ¤¸‡nµÁšnµ„´ …o°Ä—˜n°Åžœ¸Ê 1. 24 23 2. 4 3 3. 4 1 4. 24 1 25. Ĝ‹ÎµœªœÁ—È„ 12 ‡œ ¤¸Á—È„™œ´—Žoµ¥ 4 ‡œ ™oµÁ¨º°„Á—È„ 5 ‡œ ×¥„µ¦­»n¤‹µ„Á—È„Á®¨nµœ ¸Ê ¨oª‡ªµ¤ œnµ‹³Áž}œš ¸É ‹³¤¸Á—È„™œ´—Žoµ¥°¥¼nĜ„¨»n¤š ¸É Á¨º°„Ášnµ„´…o°Ä— 1. 99 35 2. 99 47 3. 99 63 4. 99 92 or248 • ло9 COMBINATORIC AND PROBABILITY>>>>>>>>>>>KRU BIRD

12 26. „¨n°ŠÄ®œ¹É Š¤¸¨¼„„oª…œµ—Á—¸¥ª„´ 3 ­¸ Áž}œ­¸…µª 4 ¨¼„ ­¸Â—ŠÂ¨³­¸Á…¸¥ª¤¸‹ÎµœªœÁšnµ„´œÁ¤ ºÉ °­»n¤ ®¥·¨¼„„oª¤µ 2 ¨¼„‡ªµ¤œnµ‹³Áž}œš ¸É‹³Å—o­¸…µªš´ÊŠ 2 ¨¼„Ášnµ„´ 15 2 ™oµ­»n¤®¥·¨¼„„oª¤µ 4 ¨¼„ ‡ªµ¤œnµ‹³Áž}œš ¸É‹³Å—o¨¼„„oªÁž}œ­¸Á…¸¥ª 1 ¨¼„ ¨³­¸Â—Š°¥nµŠœo°¥ 1 ¨¼„Ášnµ„´…o°Ä—˜n°Åžœ¸Ê 1. 70 30 2. 70 31 3. 35 29 4. 35 33 27. Ĝ„µ¦¥ºœÁ¦¸¥ŠÁž}œÂ™ª˜¦Š…°Šœ´„Á¦¸¥œµ¥ 6 ‡œ ¨³œ´„Á¦¸¥œ®·Š 4 ‡œ ™oµ„ε®œ—‡ªµ¤œnµ‹³ Áž}œš ¸ÉŤn¤¸œ´„Á¦¸¥œ®·Š­°Š‡œÄ— Ç ¥ºœ˜·—„´œÁ¨¥Ášnµ„´ a ¨³‡ªµ¤œnµ‹³Áž}œš¸Éœ´„Á¦¸¥œ®·Š š´ÊŠ®¤—˜o°Š¥ºœ˜·—„´œ Ášnµ„´ b ¨oª a + b Ášnµ„´…o°Ä—˜n°Åžœ¸Ê 1. 0.20 2. 0.25 3. 0.30 4. 0.35 28. š°—¨¼„Á˜qµ 6 ¨¼„¡¦o°¤„´œ ‡ªµ¤œnµ‹³Áž}œš ¸É ˜o¤š¸É…¹Êœ…°ŠÂ˜n¨³¨¼„‹³Å¤nŽÎʵ„´œÁ¨¥¤¸‡nµÁšnµ„´…o°Ä— 1. 324 1 2. 324 5 3. 162 1 4. 162 5 29. „¨n°ŠÄ®œ¹É Š¤¸­¨µ„®¤µ¥Á¨… 1 ™¹Š 10 ­»n¤®¥·­¨µ„¤µ 4 ĝ¡¦o°¤„´œ ‡ªµ¤œnµ‹³Áž}œš ¸É ‹³®¥· ŗoŒ¨µ„˜o¤œo°¥„ªnµ 4 ­°ŠÄ ¨³¤µ„„ªnµ 6 ®œ ¹ÉŠÄ ¤¸‡nµÁšnµ„´…o°Ä—˜n°Åžœ¸Ê 1. 35 4 2. 35 6 3. 21 1 4. 21 2 30. Ĝ„µ¦‹´—‡œ 6 ‡œ Ž¹ÉŠ¤¸œµ¥„¨³œµ¥…¦ª¤°¥¼n—oª¥Á…oµ¡´„Äœ®o°Š 3 ®o°Š ×¥š¸É®o°Šš ¸É ®œ ¹É Š¡´„ ŗo 3 ‡œ ®o°Šš ¸É ­°Š¡´„Å—o 2 ‡œ ¨³®o°Šš ¸É ­µ¤¡´„Å—o 1 ‡œ ‡ªµ¤œnµ‹³Áž}œš ¸É œµ¥„¨³œµ¥…‹³ ŗo¡´„®o°ŠÁ—¸¥ª„´œ Ášnµ„´…o°Ä—˜n°Åžœ¸Ê 1. 15 1 2. 15 3 3. 15 4 4. 15 5 ‡Îµ˜°˜´ª°¥nµŠ…o°­° Entrance Á¦ ºÉ °Š ª·›¸Á¦¸¥Š­´Áž¨¸É ¥œ‹´—®¤¼n¨³‡ªµ¤œnµ‹³Áž}œ 1. 3. 2. 1. 3. 3. 4. 3. 5. 4. 6. 3. 7. 4. 8. 3. 9. 1. 10. 3. 11. 3. 12. 2. 13. 3. 14. 2. 15. 4. 16. 4. 17. 2. 18. 4. 19. 4. 20. 3. 21. 1. 22. 2. 23. 3. 24. 1. 25. 4. 26. 2. 27. 1. 28. 2. 29. 2. 30. 3. atn249 ← лпй COMBINATORIC AND PROBABILITY>>>>>>>>>>>KRU BIRD

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