Question Number 180212 by Acem last updated on 09/Nov/22
$${How}\:{many}\:{polygons}\:{can}\:{be}\:{formed} \\ $$$$\:{from}\:{a}\:{heptagon}?\: \\ $$
Answered by mr W last updated on 10/Nov/22
$${a}\:{polygon}\:{is}\:{a}\:{closed}\:{path}\:{starting}\:{from} \\ $$$${a}\:{point}\:{and}\:{ending}\:{at}\:{the}\:{same}\:{point}. \\ $$$${let}'{s}\:{take}\:{an}\:{any}\:{point}\:{as}\:{start}\:{point} \\ $$$${and}\:{end}\:{point}.\:{to}\:{connect}\:{the}\:{other} \\ $$$$\mathrm{6}\:{points}\:{between}\:{them}\:{there}\:{are}\:\mathrm{6}!/\mathrm{2} \\ $$$${ways}.\:{that}\:{means}\:{we}\:{can}\:{form}\:\mathrm{6}!/\mathrm{2}=\mathrm{360} \\ $$$${polygons}. \\ $$$${i}\:{hope}\:{i}\:{have}\:{understood}\:{the}\:{question} \\ $$$${correctly}. \\ $$
Commented by Acem last updated on 09/Nov/22
$${Sorry},\:{i}\:{changed}\:{the}\:{question},\:{i}\:{was}\:{very}\:{drowsy} \\ $$$$\:{and}\:{unfocused}\:{when}\:{i}\:{was}\:{writing}\:{this}\:{question} \\ $$
Commented by MJS_new last updated on 09/Nov/22
$$\mathrm{no}\:\mathrm{problem},\:\mathrm{we}\:\mathrm{got}\:\mathrm{used}\:\mathrm{to}\:\mathrm{this}. \\ $$$$\mathrm{some}\:\mathrm{time}\:\mathrm{ago}\:\mathrm{another}\:\mathrm{drowsy}\:\&\:\mathrm{unfocused} \\ $$$$\mathrm{person}\:\mathrm{used}\:\mathrm{to}\:\mathrm{unwittingly}\:\mathrm{post}\:\mathrm{some}\:\mathrm{in}\:\mathrm{his} \\ $$$$\mathrm{opinion}\:\mathrm{intractable}\:\mathrm{integrals}\:\mathrm{and}\:\mathrm{each}\:\mathrm{time} \\ $$$$\mathrm{I}\:\mathrm{managed}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{one}\:\mathrm{he}\:\mathrm{was}\:\mathrm{heartbroken} \\ $$$$\mathrm{because}\:\mathrm{he}\:\mathrm{had}\:\mathrm{posted}\:\mathrm{the}\:\mathrm{wrong}\:\mathrm{question}. \\ $$$$\mathrm{better}\:\mathrm{than}\:\mathrm{the}\:\mathrm{person}\:\mathrm{who}\:\mathrm{changed}\:\mathrm{questions} \\ $$$$\mathrm{after}\:\mathrm{people}\:\mathrm{had}\:\mathrm{solved}\:\mathrm{them}\:\mathrm{to}\:\mathrm{make}\:\mathrm{them} \\ $$$$\mathrm{look}\:\mathrm{like}\:\mathrm{fools}. \\ $$
Commented by Ar Brandon last updated on 09/Nov/22
You mean Dave. Lol
Commented by Frix last updated on 09/Nov/22
$$\mathrm{Please}\:\mathrm{ignore}\:\mathrm{this}\:\mathrm{comment},\:\mathrm{I}'\mathrm{m}\:\mathrm{very}\:\mathrm{drowsy} \\ $$$$\mathrm{and}\:\mathrm{unfocused}\:\mathrm{at}\:\mathrm{the}\:\mathrm{moment}\:\mathrm{and}\:\mathrm{not}\:\mathrm{able} \\ $$$$\mathrm{to}\:\mathrm{type}\:\mathrm{at}\:\mathrm{all}. \\ $$
Commented by MJS_new last updated on 09/Nov/22
$$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{remember}\:\mathrm{the}\:\mathrm{name},\:\mathrm{but}\:\mathrm{the}\:\mathrm{weirdness} \\ $$$$\mathrm{is}\:\mathrm{rising}\:\mathrm{again}\:\mathrm{it}\:\mathrm{seems}. \\ $$
Commented by Acem last updated on 09/Nov/22
$${While}\:{am}\:{very}\:{busy}\:{with}\:{catching}\:{different}\:{ideas} \\ $$$${from}\:{my}\:{imagination}\:{to}\:{benefit}\:{those}\:{who} \\ $$$$\:{interest}\:{in}\:{science}\:{late}\:{at}\:{night},\:{there}'{s}\:{only} \\ $$$$\:{one}\:{person}\:{is}\:{very}\:{busy}\:{with}\:{catching}\:{people}, \\ $$$$\:{and}\:{he}'{s}\:{the}\:{same}\:{only}\:{one}\:{person}\:{who}\:{removes} \\ $$$$\:{his}\:{answers}\:{wich}\:{are}\:{from}\:{drowsy}\:\&\:{unfocused} \\ $$$$\:{world}\:{after}\:{seeing}\:{the}\:{logical}\:{one}\:{by}\:{the} \\ $$$$\:{honorable}\:{professors}. \\ $$$$ \\ $$$$\:{May}\:{Allah}\:{makes}\:{him}\:{very}\:{happy}\:{with}\:{his} \\ $$$$\:{hunting}. \\ $$$$ \\ $$$$\:{Anyway},\:{making}\:{people}\:{look}\:{like}\:{fools},\:{this}\:{is} \\ $$$$\:{from}\:{his}\:{own}\:{bag}. \\ $$
Commented by Acem last updated on 09/Nov/22
$$@{Mr}.{W},\:{Sir}\:{i}\:{just}\:{changed}\:{the}\:{question}\:{text},\:{so} \\ $$$$\:{as}\:{not}\:{to}\:{confuse},\:{whether}\:{we}\:{create}\:{polygons} \\ $$$$\:{from}\:{those}\:{points}\:{or}\:{we}\:{do}\:{from}\:{the}\:{vertices}\:{of} \\ $$$$\:{a}\:{heptagon},\:{so}\:{please}\:{reconsider}\:{their}\:{numbers} \\ $$$$\:{Thank}\:{you} \\ $$
Commented by mr W last updated on 10/Nov/22
$${when}\:{the}\:{question}\:{is}\:{not}\:{clear},\:{there} \\ $$$${are}\:{different}\:{ways}\:{to}\:{understand}\:{it}. \\ $$$${this}\:{is}\:{how}\:{i}\:{understand}: \\ $$$${connecting}\:{all}\:\mathrm{7}\:{points}\:{we}\:{can}\:{form} \\ $$$$\mathrm{6}!/\mathrm{2}\:{different}\:\mathrm{7}−{side}\:{polygons}. \\ $$$${certainly}\:{we}\:{can}\:{also}\:{form}\:\mathrm{6}−{side} \\ $$$${polygons},\:{there}\:{are}\:{C}_{\mathrm{6}} ^{\mathrm{7}} \mathrm{5}!/\mathrm{2}. \\ $$$${to}\:{form}\:\mathrm{3}−{side}\:{polygons},\:{there}\:{are} \\ $$$${C}_{\mathrm{3}} ^{\mathrm{7}} \mathrm{2}!/\mathrm{2}. \\ $$$${so}\:{totally}: \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left({C}_{\mathrm{7}} ^{\mathrm{7}} \mathrm{6}!+{C}_{\mathrm{6}} ^{\mathrm{7}} \mathrm{5}!+{C}_{\mathrm{5}} ^{\mathrm{7}} \mathrm{4}!+{C}_{\mathrm{4}} ^{\mathrm{7}} \mathrm{3}!+{C}_{\mathrm{3}} ^{\mathrm{7}} \mathrm{2}!\right) \\ $$$$=\mathrm{1172} \\ $$
Commented by Acem last updated on 10/Nov/22
$${Hello}\:{Sir}! \\ $$$$\mathrm{1}{st},\:{Case}\:{forming}\:{polygons}\:{from}\:{a}\:{heptagon}: \\ $$$${C}_{\mathrm{3}} ^{\mathrm{7}} +\:{C}_{\mathrm{4}} ^{\mathrm{7}} +\:{C}_{\mathrm{5}} ^{\mathrm{7}} +\:{C}_{\mathrm{6}} ^{\mathrm{7}} +\:{C}_{\mathrm{7}} ^{\mathrm{7}} \\ $$$$\mathrm{35}+\:\mathrm{35}\:+\:\mathrm{21}+\:\:\mathrm{7}\:+\:\:\mathrm{1}=\:\mathrm{99} \\ $$$$ \\ $$$${To}\:{simplify}\:{the}\:{idea},\:{suppose}\:{we}\:{want}\:{to}\:{form} \\ $$$$\:{polygons}\:{from}\:{a}\:{square}\:{C}_{\mathrm{3}} ^{\mathrm{4}} +{C}_{\mathrm{4}} ^{\mathrm{4}} \:=\:\mathrm{5} \\ $$$$ \\ $$$${Well},\:{in}\:{the}\:\mathrm{1}{st}\:{case},\:{why}\:{is}\:{there}\:{a}\:{difference} \\ $$$$\:{between}\:{us}? \\ $$$$ \\ $$
Commented by mr W last updated on 10/Nov/22
$${the}\:{difference}\:{is}\:{that}\:{you}\:{think}\:{we} \\ $$$${can}\:{only}\:{form}\:{one}\:\mathrm{4}−{sided}\:{polygon} \\ $$$${through}\:\mathrm{4}\:{points}\:{but}\:{i}\:{think}\:{we}\:{can} \\ $$$${form}\:\mathrm{3}\:\mathrm{4}−{sided}\:{polygons}\:{through}\:{the} \\ $$$$\mathrm{4}\:{points},\:{as}\:{example}. \\ $$
Commented by mr W last updated on 10/Nov/22