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

How many polygons can be formed from a heptagon?

$${How}\:{many}\:{polygons}\:{can}\:{be}\:{formed} \\ $$$$\:{from}\:{a}\:{heptagon}?\: \\ $$

Answered by mr W last updated on 10/Nov/22

a polygon is a closed path starting from a point and ending at the same point. let′s take an any point as start point and end point. to connect the other 6 points between them there are 6!/2 ways. that means we can form 6!/2=360 polygons. i hope i have understood the question correctly.

$${a}\:{polygon}\:{is}\:{a}\:{closed}\:{path}\:{starting}\:{from} \\ $$$${a}\:{point}\:{and}\:{ending}\:{at}\:{the}\:{same}\:{point}. \\ $$$${let}'{s}\:{take}\:{an}\:{any}\:{point}\:{as}\:{start}\:{point} \\ $$$${and}\:{end}\:{point}.\:{to}\:{connect}\:{the}\:{other} \\ $$$$\mathrm{6}\:{points}\:{between}\:{them}\:{there}\:{are}\:\mathrm{6}!/\mathrm{2} \\ $$$${ways}.\:{that}\:{means}\:{we}\:{can}\:{form}\:\mathrm{6}!/\mathrm{2}=\mathrm{360} \\ $$$${polygons}. \\ $$$${i}\:{hope}\:{i}\:{have}\:{understood}\:{the}\:{question} \\ $$$${correctly}. \\ $$

Commented by Acem last updated on 09/Nov/22

Sorry, i changed the question, i was very drowsy and unfocused when i was writing this question

$${Sorry},\:{i}\:{changed}\:{the}\:{question},\:{i}\:{was}\:{very}\:{drowsy} \\ $$$$\:{and}\:{unfocused}\:{when}\:{i}\:{was}\:{writing}\:{this}\:{question} \\ $$

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.

$$\mathrm{no}\:\mathrm{problem},\:\mathrm{we}\:\mathrm{got}\:\mathrm{used}\:\mathrm{to}\:\mathrm{this}. \\ $$$$\mathrm{some}\:\mathrm{time}\:\mathrm{ago}\:\mathrm{another}\:\mathrm{drowsy}\:\&\:\mathrm{unfocused} \\ $$$$\mathrm{person}\:\mathrm{used}\:\mathrm{to}\:\mathrm{unwittingly}\:\mathrm{post}\:\mathrm{some}\:\mathrm{in}\:\mathrm{his} \\ $$$$\mathrm{opinion}\:\mathrm{intractable}\:\mathrm{integrals}\:\mathrm{and}\:\mathrm{each}\:\mathrm{time} \\ $$$$\mathrm{I}\:\mathrm{managed}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{one}\:\mathrm{he}\:\mathrm{was}\:\mathrm{heartbroken} \\ $$$$\mathrm{because}\:\mathrm{he}\:\mathrm{had}\:\mathrm{posted}\:\mathrm{the}\:\mathrm{wrong}\:\mathrm{question}. \\ $$$$\mathrm{better}\:\mathrm{than}\:\mathrm{the}\:\mathrm{person}\:\mathrm{who}\:\mathrm{changed}\:\mathrm{questions} \\ $$$$\mathrm{after}\:\mathrm{people}\:\mathrm{had}\:\mathrm{solved}\:\mathrm{them}\:\mathrm{to}\:\mathrm{make}\:\mathrm{them} \\ $$$$\mathrm{look}\:\mathrm{like}\:\mathrm{fools}. \\ $$

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.

$$\mathrm{Please}\:\mathrm{ignore}\:\mathrm{this}\:\mathrm{comment},\:\mathrm{I}'\mathrm{m}\:\mathrm{very}\:\mathrm{drowsy} \\ $$$$\mathrm{and}\:\mathrm{unfocused}\:\mathrm{at}\:\mathrm{the}\:\mathrm{moment}\:\mathrm{and}\:\mathrm{not}\:\mathrm{able} \\ $$$$\mathrm{to}\:\mathrm{type}\:\mathrm{at}\:\mathrm{all}. \\ $$

Commented by MJS_new last updated on 09/Nov/22

I don′t remember the name, but the weirdness is rising again it seems.

$$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{remember}\:\mathrm{the}\:\mathrm{name},\:\mathrm{but}\:\mathrm{the}\:\mathrm{weirdness} \\ $$$$\mathrm{is}\:\mathrm{rising}\:\mathrm{again}\:\mathrm{it}\:\mathrm{seems}. \\ $$

Commented by Acem last updated on 09/Nov/22

While am very busy with catching different ideas from my imagination to benefit those who interest in science late at night, there′s only one person is very busy with catching people, and he′s the same only one person who removes his answers wich are from drowsy & unfocused world after seeing the logical one by the honorable professors. May Allah makes him very happy with his hunting. Anyway, making people look like fools, this is from his own bag.

$${While}\:{am}\:{very}\:{busy}\:{with}\:{catching}\:{different}\:{ideas} \\ $$$${from}\:{my}\:{imagination}\:{to}\:{benefit}\:{those}\:{who} \\ $$$$\:{interest}\:{in}\:{science}\:{late}\:{at}\:{night},\:{there}'{s}\:{only} \\ $$$$\:{one}\:{person}\:{is}\:{very}\:{busy}\:{with}\:{catching}\:{people}, \\ $$$$\:{and}\:{he}'{s}\:{the}\:{same}\:{only}\:{one}\:{person}\:{who}\:{removes} \\ $$$$\:{his}\:{answers}\:{wich}\:{are}\:{from}\:{drowsy}\:\&\:{unfocused} \\ $$$$\:{world}\:{after}\:{seeing}\:{the}\:{logical}\:{one}\:{by}\:{the} \\ $$$$\:{honorable}\:{professors}. \\ $$$$ \\ $$$$\:{May}\:{Allah}\:{makes}\:{him}\:{very}\:{happy}\:{with}\:{his} \\ $$$$\:{hunting}. \\ $$$$ \\ $$$$\:{Anyway},\:{making}\:{people}\:{look}\:{like}\:{fools},\:{this}\:{is} \\ $$$$\:{from}\:{his}\:{own}\:{bag}. \\ $$

Commented by Acem last updated on 09/Nov/22

@Mr.W, Sir i just changed the question text, so as not to confuse, whether we create polygons from those points or we do from the vertices of a heptagon, so please reconsider their numbers Thank you

$$@{Mr}.{W},\:{Sir}\:{i}\:{just}\:{changed}\:{the}\:{question}\:{text},\:{so} \\ $$$$\:{as}\:{not}\:{to}\:{confuse},\:{whether}\:{we}\:{create}\:{polygons} \\ $$$$\:{from}\:{those}\:{points}\:{or}\:{we}\:{do}\:{from}\:{the}\:{vertices}\:{of} \\ $$$$\:{a}\:{heptagon},\:{so}\:{please}\:{reconsider}\:{their}\:{numbers} \\ $$$$\:{Thank}\:{you} \\ $$

Commented by mr W last updated on 10/Nov/22

when the question is not clear, there are different ways to understand it. this is how i understand: connecting all 7 points we can form 6!/2 different 7−side polygons. certainly we can also form 6−side polygons, there are C_6 ^7 5!/2. to form 3−side polygons, there are C_3 ^7 2!/2. so totally: (1/2)(C_7 ^7 6!+C_6 ^7 5!+C_5 ^7 4!+C_4 ^7 3!+C_3 ^7 2!) =1172

$${when}\:{the}\:{question}\:{is}\:{not}\:{clear},\:{there} \\ $$$${are}\:{different}\:{ways}\:{to}\:{understand}\:{it}. \\ $$$${this}\:{is}\:{how}\:{i}\:{understand}: \\ $$$${connecting}\:{all}\:\mathrm{7}\:{points}\:{we}\:{can}\:{form} \\ $$$$\mathrm{6}!/\mathrm{2}\:{different}\:\mathrm{7}−{side}\:{polygons}. \\ $$$${certainly}\:{we}\:{can}\:{also}\:{form}\:\mathrm{6}−{side} \\ $$$${polygons},\:{there}\:{are}\:{C}_{\mathrm{6}} ^{\mathrm{7}} \mathrm{5}!/\mathrm{2}. \\ $$$${to}\:{form}\:\mathrm{3}−{side}\:{polygons},\:{there}\:{are} \\ $$$${C}_{\mathrm{3}} ^{\mathrm{7}} \mathrm{2}!/\mathrm{2}. \\ $$$${so}\:{totally}: \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left({C}_{\mathrm{7}} ^{\mathrm{7}} \mathrm{6}!+{C}_{\mathrm{6}} ^{\mathrm{7}} \mathrm{5}!+{C}_{\mathrm{5}} ^{\mathrm{7}} \mathrm{4}!+{C}_{\mathrm{4}} ^{\mathrm{7}} \mathrm{3}!+{C}_{\mathrm{3}} ^{\mathrm{7}} \mathrm{2}!\right) \\ $$$$=\mathrm{1172} \\ $$

Commented by Acem last updated on 10/Nov/22

Hello Sir! 1st, Case forming polygons from a heptagon: C_3 ^7 + C_4 ^7 + C_5 ^7 + C_6 ^7 + C_7 ^7 35+ 35 + 21+ 7 + 1= 99 To simplify the idea, suppose we want to form polygons from a square C_3 ^4 +C_4 ^4 = 5 Well, in the 1st case, why is there a difference between us?

$${Hello}\:{Sir}! \\ $$$$\mathrm{1}{st},\:{Case}\:{forming}\:{polygons}\:{from}\:{a}\:{heptagon}: \\ $$$${C}_{\mathrm{3}} ^{\mathrm{7}} +\:{C}_{\mathrm{4}} ^{\mathrm{7}} +\:{C}_{\mathrm{5}} ^{\mathrm{7}} +\:{C}_{\mathrm{6}} ^{\mathrm{7}} +\:{C}_{\mathrm{7}} ^{\mathrm{7}} \\ $$$$\mathrm{35}+\:\mathrm{35}\:+\:\mathrm{21}+\:\:\mathrm{7}\:+\:\:\mathrm{1}=\:\mathrm{99} \\ $$$$ \\ $$$${To}\:{simplify}\:{the}\:{idea},\:{suppose}\:{we}\:{want}\:{to}\:{form} \\ $$$$\:{polygons}\:{from}\:{a}\:{square}\:{C}_{\mathrm{3}} ^{\mathrm{4}} +{C}_{\mathrm{4}} ^{\mathrm{4}} \:=\:\mathrm{5} \\ $$$$ \\ $$$${Well},\:{in}\:{the}\:\mathrm{1}{st}\:{case},\:{why}\:{is}\:{there}\:{a}\:{difference} \\ $$$$\:{between}\:{us}? \\ $$$$ \\ $$

Commented by mr W last updated on 10/Nov/22

the difference is that you think we can only form one 4−sided polygon through 4 points but i think we can form 3 4−sided polygons through the 4 points, as example.

$${the}\:{difference}\:{is}\:{that}\:{you}\:{think}\:{we} \\ $$$${can}\:{only}\:{form}\:{one}\:\mathrm{4}−{sided}\:{polygon} \\ $$$${through}\:\mathrm{4}\:{points}\:{but}\:{i}\:{think}\:{we}\:{can} \\ $$$${form}\:\mathrm{3}\:\mathrm{4}−{sided}\:{polygons}\:{through}\:{the} \\ $$$$\mathrm{4}\:{points},\:{as}\:{example}. \\ $$

Commented by mr W last updated on 10/Nov/22

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