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SMC2012_web_solutions

L s o t h a t 2 3 4 5 6 7 8 3 5 s i n c e t h e n u m

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l s o t h a t 2 + 3 + 4 + 5 + 6 + 7 + 8 = 3 5 . S i n c e t h e n u m b e r s i n t h e r o w a n d t h o s e i n t h e c o l u m n h a v e s u m 2 1 , w e d e d u c e t h a t . x x = 2 × 2 1 - 3 5 = 7 W e n o w n e e d t w o d i s j o i n t s e t s o f t h r e e n u m b e r s c h o s e n f r o m 2 , 3 , 4 , 5 , 6 , 8 x s o t h a t t h e n u m b e r s i n b o t h s e t s t o t a l 1 4 . T h e o n l y p o s s i b i l i t i e s a r e { 2 , 4 , 8 } a n d { 3 , 5 , 6 } . W e h a v e s i x c h o i c e s o f w h i c h n u m b e r t o p u t i n t h e t o p s p a c e i n t h e v e r t i c a l l i n e , t h e n t w o f o r t h e n e x t s p a c e d o w n a n d o n e f o r t h e b o t t o m s p a c e . T h a t l e a v e s t h r e e c h o i c e s f o r t h e f i r s t s p a c e i n t h e h o r i z o n t a l l i n e , t w o f o r t h e n e x t s p a c e a n d o n e f o r t h e f i n a l s p a c e . S o t h e t o t a l n u m b e r o f w a y s i s . 6 × 2 × 1 × 3 × 2 × 1 = 7 2
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