Richard A.Brualdi 组合数学习题解答.pdf

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10 10 + C 2 10 + ··· + C 10 ; ": 2.9.h ` 10R℄|85 60 ( 6vaiBD)\BwgD 1 Sa_D. (.7bd)d 6;7bd[|dB 10a= dBB | Y = {y1, y2,··· , y10} ` 10d 6=dW>W> Yd kb ; C k X adb; C 1 10 = 210 − 1}^d; S = {s1, s2,··· , s1023},=[_ 3G=9 S|w.7bd si = sj,4 6; si; sjd |bdXdbd Y_d. 6;bd xD 9SzA7kdbd7WA2 (2.1)d40b S| .BGbX Y|dbd2 (2.1) 52+53+···+60 = 504 |G S|dB 29 − 1 = 511|b S|w.B7bd}d5?F !/a S|dk 2.10.STÆ n > 1|T.RTdÆ|bBXd> (e FR/T\7>) Sd n: ^j id>B niX 0 6 ni 6 n − 14V d>B 0,XwE n − 1d>Bd 1, 2,··· , n − 2X=9!. d>B7bd 2.16. 100dhB (a7 0)>STdT 3 bBd>B SS5?F[! 1 6 si 6 600, i = 1, 2,··· , 1023 {1, 2,··· , n} (2.1) 3

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{1, 2,··· , 6} 6 (1, j1), (2, j2),··· , (6, j6) 1; 9> 3.16. 6d:T 6 × 6xk0.:aG#Udyz4 7 2D: 4:ZGyB7z | 6v:dy} w| j1, j2,··· , j67 di2X 6!}3GjdyB 6!xDj}~9yXB~ 6!}R 6y|D 2y: yD:XdyB 6!C 2 } 3.17. 2D: 4:T 8 × 8xk0.:G#Udz | 6v:dy xD i1,··· , i6v> 8|~D 6Æ_ViY i1 < i2 < ··· < i6xD2 j1,··· , j6wi_ } d2y| X P 6 9 6yFR|D 2y:yD: C 6 } 3.18. 8:w| 5D:3: i) 8:T 8 × 8xk0.:G#Udz ii) 8:T 12 × 12xk0.:G#Udz | i)[ 16dj}z 8!C 5 } ii)[ 17dz C 8 } 3.19. d!i2dBw| 0; 9TxR (Y3℄ 0; 9TxRd! i2dB). d!i2dB 9!w 0; 9TxRd℄B~9 0 |~> ; 9(TxR)F}d 8ywEd 8BÆi2d2 8!} (0d > 0; 9di2.G)3GE_d!i2 9! − 8!} 3.20. 5d:T 8 × 8xk0:a#Ud:j;j2a 7dz (i1, j1), (i2, j2),··· , (i6, j6) {0, 1, 2,··· , 9} {0, 1,··· , 9} 8 8 P 6 8 C 2 6 8 12P 8 12C 5 8 {1, 2,··· , 8} 5

6 7 P 4 7 6 P 3 6 6 P 3 6 7 P 4 7 + 72C 3 6 P 3 6 EXbS `d |.}> :~>j:dX Ej;j2Fxk= 7 × 7dG 7v #U:XydB C 4 w>Tj;j2E0jv:dd.v:dy 7}W.v:G#UXE} 3v:℄DT 6 × 6dxkX 3|TG} 5v:dyB 72C 3 w= #U:dyB C 3 } C 4 3.21.=TÆ#b 9x 7xdV;"^R 16 x i)xRFazbd= ii)4TRb 4x 3x1 dxTD} (|RB>8) Xz_bd= (Y3℄0:D}dxd=dBX)? | i)=DRTd E dxB 9F dxB 7d;k97 16XRTd ET T |bd3GR_vFd .}W ddvk97 16 216_bd= ii)D}dx7 3 × 5d3GzbTD}d=BX 28 R 216 − 28_bd= 3.22.8 S7~B n1, n2,··· , nkdz~Wn1 = 1.8 n = n1 + n2 + ··· + nkSSd! i2BfD SS|d7 n + 1,= 3.4.2i2B RE n + 1 n1 = 1b Sd!i2B 3.23.:d!i2d2 5\45e; 1_i4\4 T\4"e4Te4"ZGz}i n1!n2!··· nk! n2!n3!··· nk! n2!··· nk! n! n! (n + 1)!

7 t2, t3, t4, t1, t6,··· ··· , t5, t7, t8 |~Ty}FÆE f \e℄ (!℄) jy\TeX 2}d\edy}0FRT\4d yRie4bX 5!}3G[i2 2 · (5!)2} 3.24.T 15v2dd~|9d 3vRb}4; 6d 3 v_gZd4Rb}4 dZv7bd|U_ZdZ v7bdZG~d.z47bd8~az} bdz4 |:~x[Xd0GFS G 15v^ t1, t2,··· , t15}.z4 ; 7bdz4w|2ÆdTiY t27jTt37j}TG! k | ; 7.bdz4~ 3vi_}4 dy P 3 }iwRE }d 12vi_ZdyX P 3 }bdz4 }i3G P 3 3.25.z~W d 11− i2dB |S| 12w 11i2| a, b, c d7 a,bT cWA ^b_℄B 3.26.z~W t2, t3, t4, t6, t1,··· ··· , t5, t7, t8 t2, t3, t4, t1, t6,··· ··· , t5, t7, t8 t3, t2, t4, t1, t6,··· ··· , t5, t7, t8 S = {3 · a, 4 · b, 5 · c} 11! 1 2!4!5! + 15 15P 3 12 ··· ··· 12 1 3!3!5! + 1 3!4!4! S = {3 · a, 4 · b, 5 · c}

EXbS `d + 1 1 + + 1 1 8 + 1 2!3!3!3! + 1 2!4!4! 3!3!5! 3!4!3! 2!3!5! 3!3!4! 10! 1 1!4!5! S = {2 · a, 1 · b, 3 · c} S = {3 · a, 3 · b, 3 · c, 3 · d} d 10− i2dB. |;[bdzb Sd 10-i2dB 3.27.z~W d 11− i2dB |4 · 11! · 3.28.2Dz~W >; 4− d d 3− >. |3-> 4-> 3.29. k}bx\Sd~B n1, n2, ··· , nkdz~Wd ( dB |T>|j i}x\daB 0, 1, 2,··· , ni ni + 13Gdd >B (n1 + 1)(n2 + 1)··· (nk + 1) 3.30.RDD 6b}!debBz}baCz (=)bT d4TT|}bwCbBaCzT d 12>3G |TdB 12j℄Dz~W S = {∞ · a1,∞ · a2,∞ · a6} bTd7 C 1 j}d℄BF? diB|d℄B7bd8 yi = xi − 1, i = 1, 2,··· , 6X? |bG aC C 6 {a, a, b, c},{a, a, c, c},{a, b, c, c},{a, c, c, c},{b, c, c, c} 1 x1 + x2 + ··· + x6 = 12 1 x1 > 1,··· , x6 > 1 {a, a, b},{a, a, c},{a, b, c},{a, c, c},{b, c, c},{c, c, c} 1 y1 + y2 + ··· + y6 = 6 1 y1 > 0,··· , y6 > 0 } 12+6−12 = C 12 17 = C 5 17 11

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