## mathmath333 one year ago Find the number of triangles formed by the Red dots ?

1. mathmath333

2. mathmath333

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3. mathmath333

\large \color{black}{\begin{align} & \normalsize \text{Find the number of triangles formed by the }\hspace{.33em}\\~\\ & \normalsize \color{red}{Red}\ \text{dots ? }\hspace{.33em}\\~\\ \end{align}}

4. mathmath333

is this correct $$\Large {\dbinom{5}{1}*\dbinom{6}{2}+\dbinom{6}{1}*\dbinom{5}{2} \\ +\dbinom{3}{2}*\dbinom{6}{1}+\dbinom{6}{2}*\dbinom{3}{1} \\ +\dbinom{3}{1}*\dbinom{5}{2}+\dbinom{5}{1}*\dbinom{3}{2} =243}$$

5. ganeshie8

$$3+5+6 = 14$$ number of triangles = $$\dbinom{14}{3} - \dbinom{3}{3} -\dbinom{5}{3}-\dbinom{6}{3}$$

6. anonymous

I would like to see all the lines drawn out please? Thank you very much. :P (jk)

7. mathmath333

i didnt understand what u did

8. ganeshie8

how many red dots are there ?

9. mathmath333

14

10. ganeshie8

how many dots make one triangle ?

11. mathmath333

3

12. ganeshie8

so, at first we might think that total possible triangles is $$\dbinom{14}{3}$$

13. ganeshie8

but immediately we see that not all of them are proper triangles

14. mathmath333

ok i get it now

15. ganeshie8

there will be some "red dot groups" that don't make a triangle what are they ?

16. mathmath333

colinear :D

17. anonymous

Oh Im slowly seeing it now. Keep going!

18. ganeshie8

BINGO! you got it!

19. mathmath333

20. anonymous

OH I GET IT NOW HAHA

21. ganeshie8

maybe try explaining it to Jhanny and us so that you understand it more...

22. mathmath333

$${\dbinom{5}{1}*\dbinom{6}{2}+\dbinom{6}{1}*\dbinom{5}{2} \\ +\dbinom{3}{2}*\dbinom{6}{1}+\dbinom{6}{2}*\dbinom{3}{1} \\ +\dbinom{3}{1}*\dbinom{5}{2}+\dbinom{5}{1}*\dbinom{3}{2} =243}$$ what was the fault in this method

23. mathmath333

I hope jhannybean has no doubt ?

24. anonymous

The 3, 5 , 6 correspond to the red dots per side, they = 14. But since each side with a specific amount of red dots that dont create a triangle, we subtract the possibilities of each side from the ones that do.

25. ganeshie8

Exactly! thats very well put!

26. mathmath333

yea she said "OH I GET IT NOW HAHA" she has not doubt .

27. anonymous

Haha, Im a litle slow.

28. ganeshie8

could you explain ur method

29. mathmath333

to make triangle we need 1 red dot from one line and 2 red dots from other lines vice versa. count all 3 lines as such +vice versa

30. ganeshie8

Looks you forgot to count the case : 1 red dot from each line

31. mathmath333

but i forgot the the case 1 dot from all 3 lines i got my mistake

32. ganeshie8

good

33. mathmath333

thanks

34. mathmath333

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35. imqwerty

kewl cx

36. mathmath333

profile pictures dont support gif images

37. imqwerty

yea :)

38. mathmath333

by the way what is 'kewl'

39. imqwerty

( ͡° ͜ʖ ͡°) *cool