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Question Number 67281 by Tony Lin last updated on 25/Aug/19

Commented by mr W last updated on 25/Aug/19

$I g e t 3 C_{2}^{5} + 3 C_{2}^{5} + 12 = 72$
Commented by Tony Lin last updated on 25/Aug/19

$R e p o s t Q u e s t i o n N o . 66681$
Commented by Tony Lin last updated on 25/Aug/19

$C_{2}^{3} C_{2}^{5} \times 3 - C_{2}^{3} C_{2}^{3} \times 2 = 72$
Commented by Tony Lin last updated on 25/Aug/19

Commented by Tony Lin last updated on 25/Aug/19

$o r a n g e p a r t (2 \times 4 i n s i d e)$ $\Rightarrow c h o o s e t w o f r o m t h r e e v e r t i c a l l i n e s$ $\Rightarrow c h o o s e t w o f r o m f i v e h o r i z o n t a l l i n e s$ $\Rightarrow C_{2}^{3} C_{2}^{5} = 30$ $b l u e p a r t (2 \times 4 o u t s i d e)$ $\Rightarrow t h e s a m e a s t h e o r a n g e p a r t$ $\Rightarrow C_{2}^{3} C_{2}^{5} = 30$ $g r e e n p a r t (2 \times 4 m i d d l e)$ $\Rightarrow o n l y i n t h i s p a r t t h e r e c t a n g l e s o u t s i d e$ $c o m b i n e w i t h t h e r e c t a n g l e s i n s i d e$ $\Rightarrow t h e s a m e a s t h e o r a n g e p a r t$ $\Rightarrow C_{2}^{3} C_{2}^{5} = 30$ $p i n k p a r t (2 \times 2 i n s i d e)$ $\Rightarrow w e h a v e a l r e a d y c o u n t e d t h i s p a r t$ $i n t h e o r a n g e p a r t b u t w e c o u n t t h i s$ $p a r t a g a i n i n t h e g r e e n p a r t$ $\Rightarrow c h o o s e t w o f r o m t h r e e v e r t i c a l l i n e s$ $\Rightarrow c h o o s e t w o f r o m t h r e e h o r i z o n t a l l i n e s$ $\Rightarrow C_{2}^{3} C_{2}^{3} = 9$ $y e l l o w p a r t (2 \times 2 o u t s i d e)$ $\Rightarrow w e h a v e a l r e a d y c o u n t e d t h i s p a r t$ $i n t h e b l u e p a r t b u t w e c o u n t t h i s$ $p a r t a g a i n i n t h e g r e e n p a r t$ $\Rightarrow t h e s a m e a s t h e p i n k p a r t$ $\Rightarrow C_{2}^{3} C_{2}^{3} = 9$ $∴ t h e r e a r e (C_{2}^{5} C_{2}^{3} \times 3 - C_{2}^{3} C_{2}^{3} \times 2 = 72) r e c t a n g l e s$
Commented by Rasheed.Sindhi last updated on 25/Aug/19

$Sir Tony Lin, I {don}^{'} t see any error in$ $your approach . However I want to see$ ${where}^{'} s deffect in my approach! Pl$ $cooperate with me .$
	Answered by Rasheed.Sindhi last updated on 25/Aug/19

	Commented by Rasheed.Sindhi last updated on 25/Aug/19

	$^{∙} I n o r d e r t o p r e v e n t m i s s i n g / r e p e a t i n g$ $i n c o u n t i n g, {w e}^{'} l l c o u n t r e c t a n g l e s w i t h$ $t h e i r l e f t - u p p e r c o r n e r s . F o r e x a m p l e$ $a l l t h e r e c t a n g l e s w h o s e l e f t - u p p e r$ $c o r n e r i s A w i l l b e c o u n t e d t o g e t h e r$ $a n d r e c o r d e d w i t h A . I n t h i s w a y w e$ $w i l l c o u n t r e c t a n g l e s w i t h e a c h a n d$ $e v e r y v e r t e x .$ $^{∙} A : E d g e s f r o m A t o t h e l e f t :$ $AB, AC (2 e d g e s)$ $E d g e s f r o m A t o t h e d o w n :$ $AG, AL, AQ, {AA}_{1} (4 e d g e s)$ $∴ N u m b e r o f r e c t a n g l e s w i t h l e f t -$ $u p p e r c o r n e r A = 2 \times 4 = 8$ $(R e c a l l t h a t e v e r y u p p e r e d g e$ $m a k e s a r e c t a n g l e w i t h e v e r y$ $l e f t e d g e .)$ $. . . . . . .$
	Commented by Tony Lin last updated on 25/Aug/19
	$A:(AB,AC)×(AG,AL,AQ,AA_1 )=8 B:BC×(BI,BN,BS,BB_1 )=4 C:0 D:(DE,DF)×(DH,DM,DR,DV)=8 E:EF×(EI,EN,ES,EW)=4 F:0 G: { (((GI,GK)×(GL,GQ,GA_1 )=6)),(((GH,GJ)×(GL,GQ)=4)) :} H: { (((HI,HJ)×(HM,HR,HV)=6)),((HK×(HM,HR)=2)) :} I: { ((IJ×(IN,IS,IW)=3)),((IK×(IN,IS,IW)=3)) :} J:JK×(JO,JT)=2 K:0 L: { (((LN,LP)×(LQ,LA_1 )=4)),(((LM,LO)×LQ=2)) :} M: { (((MN,MO,MP)×MR=3)),(((MN,MO)×MV=2)) :} N: { ((NO×(NS,NW)=2)),((NP×(NS,NB_1 )=2)) :} O:OP×OT=1 P:0 Q:(QS,QU)×QA_1 =2 R:(RS,RT)×RV=2 S: { ((ST×SW=1)),((SU×SB_1 =1)) :} T∼C_1 :0 SUM:72 a little complex, but the result is the same this is the power of classification$
	$A : (A B, A C) \times (A G, A L, A Q, {A A}_{1}) = 8$ $B : B C \times (B I, B N, B S, {B B}_{1}) = 4$ $C : 0$ $D : (D E, D F) \times (D H, D M, D R, D V) = 8$ $E : E F \times (E I, E N, E S, E W) = 4$ $F : 0$ $G : {\begin{cases} (G I, G K) \times (G L, G Q, {G A}_{1}) = 6 \\ (G H, G J) \times (G L, G Q) = 4 \end{cases}$ $H : {\begin{cases} (H I, H J) \times (H M, H R, H V) = 6 \\ H K \times (H M, H R) = 2 \end{cases}$ $I : {\begin{cases} I J \times (I N, I S, I W) = 3 \\ I K \times (I N, I S, I W) = 3 \end{cases}$ $J : J K \times (J O, J T) = 2$ $K : 0$ $L : {\begin{cases} (L N, L P) \times (L Q, {L A}_{1}) = 4 \\ (L M, L O) \times L Q = 2 \end{cases}$ $M : {\begin{cases} (M N, M O, M P) \times M R = 3 \\ (M N, M O) \times M V = 2 \end{cases}$ $N : {\begin{cases} N O \times (N S, N W) = 2 \\ N P \times (N S, {N B}_{1}) = 2 \end{cases}$ $O : O P \times O T = 1$ $P : 0$ $Q : (Q S, Q U) \times {Q A}_{1} = 2$ $R : (R S, R T) \times R V = 2$ $S : {\begin{cases} S T \times S W = 1 \\ S U \times {S B}_{1} = 1 \end{cases}$ $T \sim C_{1} : 0$ $S U M : 72$ $a l i t t l e c o m p l e x, b u t t h e r e s u l t i s t h e s a m e$ $t h i s i s t h e p o w e r o f c l a s s i f i c a t i o n$
	Commented by peter frank last updated on 25/Aug/19

	$t h a n k y o u b o t h$
	Commented by Rasheed.Sindhi last updated on 25/Aug/19

	$G r e a t T h a n k s f o r G r e a t H e l p S \overset{⧫}{∥} R!$ $I f o u n d m y m i s t a k e a f t e r c o m p a r i n g$ $m y w o r k w i t h y o u r s!$ $A l s o t h a n k ⋓ P e t e r F r a n k!$
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