Maximum number of lines drawn out of n points

There are n distinct points in a plane. No three of the points are collinear. Find the maximum number of distinct lines that can be drawn if these points are connected. Express your answer in fraction form. (Use LaTeX syntax, enclose your answer in dollar signs ($): $\frac{1}{2}$)

Answer:

(Based on Mathcounts National 1990 Sprint Problem 29)

Correct! The answer is $\binom n2=\frac{n(n-1)}{2}$

Incorrect! Try again.

import simplegui points = [(20,80),(60,40),(140,100),(250,25),(155,145)] pointsIds = [(20,72),(60,35),(140,95),(254,30),(155,158)] messageNumPoints = "n=3" messageNumLines="3 lines" n=3 # Handler for buttons def increasen(): global messageNumPoints ,messageNumLines,n if (n<6): n +=1 messageNumPoints = "n=" + str(n) messageNumLines = str(n*(n-1)/2) + " lines" def decreasen(): global messageNumPoints ,messageNumLines,n if (n>3): n -=1 messageNumPoints = "n=" + str(n) messageNumLines = str(n*(n-1)/2) + " lines" def draw(canvas): if n>5: messageNumLines1 = "For a general n, point A can connect " messageNumLines2 = "to at most n-1 points thus forming n-1 " messageNumLines3 = "lines. Point B can connect to at most " messageNumLines4 = "n-2 points forming n-2 lines..." messageNumLines5 = "So the answer is n-1+n-2+...+1 lines." messageNumLines6 = "Alternatively to draw a line we need to " messageNumLines7 = " pick 2 points out of the n points. " messageNumLines8 = "We have n choices for the first point and " messageNumLines9 = "n-1 choices for the second point. " messageNumLines10 = " Since each such pair is counted twice " messageNumLines11 = " we divide by 2." canvas.draw_text(messageNumLines1,[10,20],18,"Red") canvas.draw_text(messageNumLines2,[10,40],18,"Red") canvas.draw_text(messageNumLines3,[10,60],18,"Red") canvas.draw_text(messageNumLines4,[10,80],18,"Red") canvas.draw_text(messageNumLines5,[10,100],18,"Red") canvas.draw_text(messageNumLines6,[10,120],18,"Green") canvas.draw_text(messageNumLines7,[10,140],18,"Green") canvas.draw_text(messageNumLines8,[10,160],18,"Green") canvas.draw_text(messageNumLines9,[10,180],18,"Green") canvas.draw_text(messageNumLines10,[10,200],18,"Green") canvas.draw_text(messageNumLines11,[10,220],18,"Green") else: canvas.draw_text(messageNumPoints, [120,20], 28, "Red") canvas.draw_text("A", pointsIds[0], 10, "White") canvas.draw_text("B", pointsIds[1], 10, "White") canvas.draw_text("C", pointsIds[2], 10, "White") if n>3: canvas.draw_text("D", pointsIds[3], 10, "White") if n>4: canvas.draw_text("E", pointsIds[4], 10, "White") canvas.draw_circle(points[0],3,2,"Green","Green") canvas.draw_circle(points[1],3,2,"Red","Red") canvas.draw_circle(points[2],3,2,"Blue","Blue") if n>3: canvas.draw_circle(points[3],3,2,"Orange","Orange") if n>4: canvas.draw_circle(points[4],3,2,"White","White") canvas.draw_line(points[0],points[1],2,"Yellow") canvas.draw_line(points[0],points[2],2,"Yellow") canvas.draw_line(points[1],points[2],2,"Cyan") if n>3: canvas.draw_line(points[0],points[3],2,"Yellow") canvas.draw_line(points[1],points[3],2,"Cyan") canvas.draw_line(points[2],points[3],2,"Red") if n>4: canvas.draw_line(points[0],points[4],2,"Yellow") canvas.draw_line(points[1],points[4],2,"Cyan") canvas.draw_line(points[2],points[4],2,"Red") canvas.draw_line(points[3],points[4],2,"Green") canvas.draw_text(messageNumLines, [120,220], 28, "Red") # Create a frame and assign callbacks to event handlers frame = simplegui.create_frame("How many lines?", 300, 250) frame.add_button("From n=3 to n=5", increasen) frame.add_button("From n=5 to n=3", decreasen) frame.set_draw_handler(draw) # Start the frame animation frame.start()