There are two parallel lines having 4 and 6 points on them. How many triangles can be formed from these vertices?
L1 has 4 points and L2 has 6 points. pick two from L1 (4C2) and one from L2(6C1) to form a triangle count= 6*6=36 triangles pick two from L2(6C2) and one from L1(4C1) count=15*4=60 triangles so total triangles=36+60=96 triangles.
Is it possible for a knight in a game of chess to cover all the places without repeating any of them and return to its original position?
After every turn knight lands on a position which has a different color than its previous position. So we can divide the board into bipartite sets of white and black. knight always travels from white to black and black to white. If knight starts on black box then after even moves, it will still be on black box. After 63 moves knight will reach the final box and it will be on white color. So the 64th move will get it back to its original black box and hence it is possible.
(7x-2)^(1/3) + (7x+5)^(1/3) = 3 solve for x.
assuming s= 7x-2 equation becomes : s^(1/3) + (s+7)^(1/3)=3 => (s+7)^(1/3)=3-s^(1/3) now cubing both sides, we get, s+7=27-s-9*s^(1/3)(3-s^1/3) now taking t=s^1/3, we get, 2t^3 - 9t^2 +27t -20=0 t=1 is a solution to this equation s=t^3=1 x=(s+2)/7=3/7