How many polygons can be formed from a heptagon?


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Question Number 180212 by Acem last updated on 09/Nov/22

$H o w m a n y p o l y g o n s c a n b e f o r m e d$ $f r o m a h e p t a g o n ?$
	Answered by mr W last updated on 10/Nov/22

	$a p o l y g o n i s a c l o s e d p a t h s t a r t i n g f r o m$ $a p o i n t a n d e n d i n g a t t h e s a m e p o i n t .$ ${l e t}^{'} s t a k e a n a n y p o i n t a s s t a r t p o i n t$ $a n d e n d p o i n t . t o c o n n e c t t h e o t h e r$ $6 p o i n t s b e t w e e n t h e m t h e r e a r e 6! / 2$ $w a y s . t h a t m e a n s w e c a n f o r m 6! / 2 = 360$ $p o l y g o n s .$ $i h o p e i h a v e u n d e r s t o o d t h e q u e s t i o n$ $c o r r e c t l y .$
	Commented by Acem last updated on 09/Nov/22

	$S o r r y, i c h a n g e d t h e q u e s t i o n, i w a s v e r y d r o w s y$ $a n d u n f o c u s e d w h e n i w a s w r i t i n g t h i s q u e s t i o n$
	Commented by MJS_new last updated on 09/Nov/22

	$no problem, we got used to this .$ $some time ago another drowsy & unfocused$ $person used to unwittingly post some in his$ $opinion intractable integrals and each time$ $I managed to solve one he was heartbroken$ $because he had posted the wrong question .$ $better than the person who changed questions$ $after people had solved them to make them$ $look like fools .$
	Commented by Ar Brandon last updated on 09/Nov/22
	You mean Dave. Lol
	Commented by Frix last updated on 09/Nov/22

	$Please ignore this comment, I^{'} m very drowsy$ $and unfocused at the moment and not able$ $to type at all .$
	Commented by MJS_new last updated on 09/Nov/22

	$I {don}^{'} t remember the name, but the weirdness$ $is rising again it seems .$
	Commented by Acem last updated on 09/Nov/22

	$W h i l e a m v e r y b u s y w i t h c a t c h i n g d i f f e r e n t i d e a s$ $f r o m m y i m a g i n a t i o n t o b e n e f i t t h o s e w h o$ $i n t e r e s t i n s c i e n c e l a t e a t n i g h t, {t h e r e}^{'} s o n l y$ $o n e p e r s o n i s v e r y b u s y w i t h c a t c h i n g p e o p l e,$ $a n d {h e}^{'} s t h e s a m e o n l y o n e p e r s o n w h o r e m o v e s$ $h i s a n s w e r s w i c h a r e f r o m d r o w s y & u n f o c u s e d$ $w o r l d a f t e r s e e i n g t h e l o g i c a l o n e b y t h e$ $h o n o r a b l e p r o f e s s o r s .$ $M a y A l l a h m a k e s h i m v e r y h a p p y w i t h h i s$ $h u n t i n g .$ $A n y w a y, m a k i n g p e o p l e l o o k l i k e f o o l s, t h i s i s$ $f r o m h i s o w n b a g .$
	Commented by Acem last updated on 09/Nov/22

	$@ M r . W, S i r i j u s t c h a n g e d t h e q u e s t i o n t e x t, s o$ $a s n o t t o c o n f u s e, w h e t h e r w e c r e a t e p o l y g o n s$ $f r o m t h o s e p o i n t s o r w e d o f r o m t h e v e r t i c e s o f$ $a h e p t a g o n, s o p l e a s e r e c o n s i d e r t h e i r n u m b e r s$ $T h a n k y o u$
	Commented by mr W last updated on 10/Nov/22

	$w h e n t h e q u e s t i o n i s n o t c l e a r, t h e r e$ $a r e d i f f e r e n t w a y s t o u n d e r s t a n d i t .$ $t h i s i s h o w i u n d e r s t a n d :$ $c o n n e c t i n g a l l 7 p o i n t s w e c a n f o r m$ $6! / 2 d i f f e r e n t 7 - s i d e p o l y g o n s .$ $c e r t a i n l y w e c a n a l s o f o r m 6 - s i d e$ $p o l y g o n s, t h e r e a r e C_{6}^{7} 5! / 2 .$ $t o f o r m 3 - s i d e p o l y g o n s, t h e r e a r e$ $C_{3}^{7} 2! / 2 .$ $s o t o t a l l y :$ $\frac{1}{2} (C_{7}^{7} 6! + C_{6}^{7} 5! + C_{5}^{7} 4! + C_{4}^{7} 3! + C_{3}^{7} 2!)$ $= 1172$
	Commented by Acem last updated on 10/Nov/22

	$H e l l o S i r!$ $1 s t, C a s e f o r m i n g p o l y g o n s f r o m a h e p t a g o n :$ $C_{3}^{7} + C_{4}^{7} + C_{5}^{7} + C_{6}^{7} + C_{7}^{7}$ $35 + 35 + 21 + 7 + 1 = 99$ $T o s i m p l i f y t h e i d e a, s u p p o s e w e w a n t t o f o r m$ $p o l y g o n s f r o m a s q u a r e C_{3}^{4} + C_{4}^{4} = 5$ $W e l l, i n t h e 1 s t c a s e, w h y i s t h e r e a d i f f e r e n c e$ $b e t w e e n u s ?$
	Commented by mr W last updated on 10/Nov/22

	$t h e d i f f e r e n c e i s t h a t y o u t h i n k w e$ $c a n o n l y f o r m o n e 4 - s i d e d p o l y g o n$ $t h r o u g h 4 p o i n t s b u t i t h i n k w e c a n$ $f o r m 3 4 - s i d e d p o l y g o n s t h r o u g h t h e$ $4 p o i n t s, a s e x a m p l e .$
	Commented by mr W last updated on 10/Nov/22

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