Sunday, December 7, 2014

Problem Solving Strategies: Applications of the “Choose 2” method

1. Example: How many diagonals can be drawn for a polygon with “n” sides?

Method I:

The number of diagonals in a polygon = n(n-3)/2, where n is the number of polygon sides.

For a convex n-sided polygon, there are n vertexes, and from each vertex you can draw n-3 diagonals, so the total number of diagonals that can be drawn is n (n-3).

However, this would mean that each diagonal would be drawn twice, (to and from each vertex), so the expression must be divided by 2.

Method II:

nC2 (choose 2) - n sides = n(n-1)/ 2 – n sides

2. Example: There are n people at a party, each person shakes hands with the every other person once. How many handshakes?

Method I:
nC2 in this case (10 x 9) /2 =45
__ __ First slot you have 10 persons to choose from, second slot 9 persons. Since A shakes hands with B is the same as B shakes hands with A, so you divide the number by 2 and get the answer.

Method II:
Sum of the first consecutive Natural numbers: n (n+1) /2
The first person shakes hands with 9 other person; the second person shakes hands with 8 other person, etc…
So 9 + 8 + 7 + …= (9 x 10)/ 2 = 45

3. Example: N dots evenly spaced on a circle. How many chords can you make using those dots?

Methods: This is very similar to hand-shaking questions.

I: nC2

II: Sum of the first consecutive (N-1) Natural numbers

Word problems: Answers below.

#1 : A convex polygon with n sides has 20 diagonals. How many diagonals does an (n+1)-sided convex polygon have?

#2: A polygon has n sides and n diagonals. What is n? 

#3: How many diagonals does a decagon have?

#4: How many diagonals does a dodecagon have? 

#5: How many line segments have both their endpoints located at the vertexes of a given cube?

#6: There are 8 points on a circle, how many lines can you make? How many triangles can you make?

#7: If each of the interior angles of two regular polygons adds up to 255 degrees and their diagonals add up to 29, what is the sum of their sides?


# 1- 27    #2 -5     #3-35      #4-54 [12C2 -12]     #5 28 [There are 8 vertices, so 8C2]

#6  8C2 = 28 for lines and 8C3 = 56 triangles   

#7: 6[hexagon] + 8[octagon] = 14 sides