In a game with two players X and Y and given n (≥ 4) number of circles drawn on a plane paper, alternately each one of X and Y is suppose to join two different circles which were not already joined by a line. The winner is one who joins the last available pair of circles and the game ends. If X starts the game then:

Explanation:There can be many questions like this so INSTEAD of going with a general method of solving this problem, we are proceeding with reviewing options here,We can start with an example of n = 4,So, to connect 4 Circles we need 6 lines,which can be stated as,To connect n Circles we need nC2 lines, since one line can connect two circles.So we have here,If X starts the game with n circles,nC2 must be ODD so that X can win.nC2 must be even so that Y can win.Now, we can draw a table as,nnc246510615721828936Now from options, we can CHECK,X cannot always win since nC2 not always odd.Y cannot always win since nC2 not always even.X wins whenever n is odd is also not true since when n = 5, which is odd, X does not win.Y wins whenever n is a power of 2 is correct since when n = 4, 8, 16, etc. nC2 becomes even and Y wins.