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how many triangles are possible?

• Combination for selecting 3 points 15c3=455
  Now 3 row and 5 column that make 43 cases of not selecting
  Explaining.......for 43
  When row is taken-3 points from 5 points in one row-from each row selecting 3 points=5c3+5c3+5c3=30 cases
  When column is taken-There is three dots in a column, and there are five columns, so that makes 5 cases.
  Now see the diagonal case- 3 cases for slanted diagonal + 3 cases for opposite slanted diagonal makes - 6 cases
  Total of 43 cases
  We have to subtract it from the total number of cases for possible traingles
  
  455-43 =412
• 9 years agoHelpfull: Yes(10) No(1)
• dear friends, there is a great complexity in understanding the case 43..
  i will tell u in details.
  15c3 is total triangles possible
  (3 rows)*5c3 = 30 as they are collinear points.
  same applies for column operation (5 columns)*3c3 = 5
  now diagonals are also considered as triangle cant be formed through three points in straight line.
  if rows are a,b,c and columns are 1,2,3,4,5 then diagonals a1,b2,c3 ...a3,b4,c5 = (3) and reverse again as a5,b4,c3 ....a3,b2,c1 =(3) diagonals are there.
  now 2 bigger diagonals a1,b3,c5 and a5,b3,c1 are taken = (2).
  therefore total - non possible cases = possible triangles.
  455 - ((30)+(5)+(3)+(3)+(2)) = 412 answer.!!
• 8 years agoHelpfull: Yes(6) No(0)
• Combination for selecting 3 points 15c3=455
  Now 3 row and 5 column that make 43 cases of not selecting
  455-43 =412
  
• 9 years agoHelpfull: Yes(3) No(1)
• sorry it should be 15C2=105
• 9 years agoHelpfull: Yes(2) No(5)
• there are 15 points . So the number of triangles are 15C2=10
• 9 years agoHelpfull: Yes(1) No(3)
• 15c3 - 5c3-5c3-5c3-3c3-3c3-3c3-2*(3c3+3c3+3c3)=412
• 9 years agoHelpfull: Yes(1) No(1)
• 15C3 - 3*(5C3)= 425
  Choosing all possible combinations then subtracting the cases in which triangle was allowed to be formed by co linear points.
• 8 years agoHelpfull: Yes(1) No(0)
• 30 + 5 for row and column i got this..but for reamining how it should be possible can u please explain @chairag??
• 8 years agoHelpfull: Yes(1) No(0)
• 15C2 - 3*5C2=105-30=75
  So total 75 triangles are possible bcoz points which are in straight line will not form triangle....
• 9 years agoHelpfull: Yes(0) No(3)
• please explain more? why there are 43 cases of not selecting? @CHIRAG SHAH
• 9 years agoHelpfull: Yes(0) No(0)
• it should be 15C3-5c2-1
• 9 years agoHelpfull: Yes(0) No(0)
• I missed one point-There could be a diagonal of the form (1,1), (2, 3), (3, 5), or the one in the opposite direction, so that makes 2 cases.
  
• 9 years agoHelpfull: Yes(0) No(0)
• the diagonal cases are not clear @chirag shah??
• 9 years agoHelpfull: Yes(0) No(0)
• please explain it
• 8 years agoHelpfull: Yes(0) No(0)
• 455-57=398.., (57 because this no of cases triangle would not form)
  
• 8 years agoHelpfull: Yes(0) No(1)