No. of triangles. Geometry and Number Theory.

Geometry Level 5

Find the digit sum of the number of triangles formed by intersecting diagonals of an 8-sided figure? :D

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The answer is 11.

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3 solutions

A n e i g h t s i d e d f i g u r e c o n t a i n s 8 v e r t i c e s , n o n e o f w h i c h a r e c o l i n e a r . S o , t h e q u e s t i o n r e d u c e s t o t h e f o r m a t i o n o f t r i a n g l e s f r o m 8 p o i n t s i n a p l a n e , n o n e o f w h i c h a r e c o l i n e a r . S i n c e t h e r e a r e 8 p o i n t s , a n d t h r e e p o i n t s a r e r e q u i r e d t o c o n s t r u c t a t r i a n g l e . T h e r e f o r e , t h e n u m b e r o f w a y s i n w h i c h t h e s e 3 p o i n t s c a n b e s e l e c t e d i s g i v e n b y c o m b i n a t i o n : C ( 8 , 3 ) = 8 ! 5 ! 3 ! = 8 × 7 × 6 × 5 ! 5 ! 3 × 2 = 56 S u m o f d i g i t s = 5 + 6 = 11 An\quad eight\quad sided\quad figure\quad contains\quad 8\quad vertices,\quad none\quad of\quad \\ which\quad are\quad colinear.\quad So,\quad the\quad question\quad reduces\quad to\\ the\quad formation\quad of\quad triangles\quad from\quad 8\quad points\quad in\\ a\quad plane,\quad none\quad of\quad which\quad are\quad colinear.\\ Since\quad there\quad are\quad 8\quad points,\quad and\quad three\quad points\\ are\quad required\quad to\quad construct\quad a\quad triangle.\quad Therefore,\\ the\quad number\quad of\quad ways\quad in\quad which\quad these\quad 3\quad points\quad can\\ be\quad selected\quad is\quad given\quad by\quad combination:\\ C(8,3)=\frac { 8! }{ 5!3! } =\frac { 8\times 7\times 6\times 5! }{ 5!3\times 2 } =56\\ \\ Sum\quad of\quad digits=5+6=11\quad

CHEERS!!

This is fallacious..

Pratik Shastri - 6 years, 6 months ago
Emanuel Valente
Jul 27, 2014

A complete answer here: https://cs.uwaterloo.ca/journals/JIS/sommars/newtriangle.html

Astro Enthusiast
Jul 26, 2014

The number of triangles is 632 all in all. 6+3+2=11

what does it mean

Smayan Das - 6 years, 10 months ago

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It means that there are 632 triangles all in all. It includes the small triangles that are formed by intersecting diagonals from vertices to the other. Do I answer your question? :)

Astro Enthusiast - 6 years, 10 months ago

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# Still dont get you. :-(

Chinmay Raut - 6 years, 10 months ago

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@Chinmay Raut do you know the problems such as count the number of squares? This problem is like that. You need to count even the smallest of the triangles :)

Astro Enthusiast - 6 years, 10 months ago

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@Astro Enthusiast

Oh kk ....... [you could have told that in the first place :-P ] ty anyways :-)

Chinmay Raut - 6 years, 10 months ago

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@Chinmay Raut I'm sorry!

Astro Enthusiast - 6 years, 10 months ago

Couldn't this problem be solved by Permutation and Combination? Just guessing. Can someone answer this? @Anugrah Agrawal

Nikhil Tandon - 6 years, 10 months ago

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Hey @Nikhil Tandon I have solved it by permutations and combinations...please take a look at it as well...

A Former Brilliant Member - 6 years, 9 months ago

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