Math Problem/Process Example
Robin McAteer (PDP 4220)
Curriculum Connection
MDM4U General Strand: Solve Problems using Counting Principles
2.1 recognize the use of permutations and combinations as counting techniques with
advantages over other counting techniques (e.g., making a list; using a tree
diagram; making a chart; drawing a Venn diagram), distinguish
between situations that involve the use of permutations and those that involve the
use of combinations (e.g., by considering whether or not order matters), and make
connections between, and calculate, permutations
2.2 solve simple problems using techniques for counting permutations and
combinations, where all objects are distinct, and express the solutions using
standard combinatorial notation [e.g., n!, P(n, r), ( )]
2.5 Solve probability problems using counting principles for situations involving
equally likely outcomes
Problem Statement
A new lottery, DATA-o-RAMA, is being created, with two different options.
In option 1, the player chooses 3 different numbers between 1 and 20. In order to win, the
player must select the correct three numbers in the correct order.
In option 2, the player chooses 4 different numbers between 1 and 30.
player must select the correct numbers in any order.
In order to win, the
a) Which option gives the best chance of winning?
b) How could the games be adjusted so that the two options had similar chances of
winning?
Prerequesite Skills



Exposure to tree diagrams and the Fundamental Counting Principle
Factorial notation and calculations
Basic knowledge of permutations and combinations and formulas
Intuitively: Consider the two options. Which has fewer outcomes?
Probability of winning : Option 1
Experiment: Choosing three different numbers between 1 and 20
Sample Outcomes: 1-2-3, 13-18-3, 3-13-18
6840
branches
Only one of the branches represents the “winning” outcome,
Therefore, the probability of winning with Option 1 is
Algebraic Solution
p ( winning) 
1
6840
n ( winning)
n (outcomes)
Different permutations (arrangements)
of 3 numbers out of 20
(n=20, r=3)
P ( n, r ) 
n!
, n  r  0
(n  r )!
1

P ( 20 ,3)


1
 20! 
 17! 
 
17!
20!

Notice how this relates
to the tree diagram


17  16  ...  2  1
20  19  18  17  16  ...  2  1
1
20  19  18
1
6840
Probability of winning : Option 2
Experiment: Choosing four different numbers between 1 and 30
Sample Outcomes: 1-2-3-4, 13-18-3-20, 3-13-18-21 etc.
657 720
branches
How many of the branches represent a “winning” outcome?
Assume that the winning numbers are 1-2-3-4 in any order.
How many branches of the tree contain these 4 numbers?
Rephrased, how many ways
different ways can the 4
winning numbers be arranged?
P( 4,4)  4!  24
Therefore, the probability of
this winning is:
n( winning outcomes)
24

n(outcomes)
657720
1

27405
Systematic List of Winning Outcomes
1
1
1
1
1
1
2
2
3
3
4
4
3
4
2
4
2
3
4
3
4
2
3
2
6 branches
Similarly, there are 6 branches
(arrangements) of winning
outcomes starting with
2, 3, and 4
Therefore, the total number branches containing these
winning numbers is 6  4  24
The probability of winning Option2 is
1
27 405
Therefore, Option 1 has a much better probability of winning !
Algebraic Solutions
2. Using Combinations
1. Using Permutations
In this case, an outcome is a branch of the
tree diagram. Each different ordering of the
same numbers is considered a different outcome,
so there are several winning outcomes.
p ( winning) 
In this case, an outcome is a group of 4
different numbers. Order is not important,
so there is only one winning outcome.
p ( winning) 
n (outcomes)
n ( winning)

n (outcomes)

P ( 4, 4 )
P ( 30 ,4 )


4!
 30! 
 26! 
 
Number of different
combinations (groupings) of 4
numbers out of 30
(n=30, r=4)
n
n!
,n  r  0
  
 r  (n  r )! r!
26!4!
30!


30  29  28  27
1
27 405
Notice that
these are the
same! Why?
b) How could the games be adjusted so that
the two options had similar chances of
winning?
If we keep option 1 the same, how could we change
option 2 give 6 840 outcomes instead of 27 407 ?
Choose a smaller range of numbers (trial and error)
Range
Number of
Combinations
1-25
 25  25!
  
= 12 650
 4  21!4!
 22  22!
= 7 315
  
 4  18!4!
1-22
1-21
 21 21!
  
= 5 985
 4  17!4!



24
Closest
option
n ( winning)

1
 30 
 
 4
1
 30! 
 26!4! 


26!4!
30!
4  3 2 1
30  29  28  27
1
27 405
Algebraic Solution
substitute r =4, solve
for n so that
number of outcomes  6840
n
n!
number of outcomes    
 4  ( n  4)!4!
n!

 6 840
( n  4)!4!
n!
 (6840)( 4! )
( n  4)!
n( n  1)( n  2)( n  3)  164 160
Use Calculator
Solve by looking for intersection of :
y1  n(n  1)( n  2)( n  3) , y 2  164 160
n  21.66
Therefore, by using a range numbers from 1-22 instead of 1-30, the
probabilities are more closely matched.
Mathematical Process Look-fors / Checklist
Process
Problem Solving
Look-Fors
-
Reasoning and
Proving
-
Reflecting
-
Connecting
-
Selecting Tools
and Computation
Strategies
-
Representing
-
Communicating
-
draws a tree diagram that accurately represents the
problem, and uses it to solve the problem
Uses other problem solving strategies (lists, tables) to
identify patterns
Tries a variety of strategies in part b)
Uses guess and check in part b)
makes a reasoned guess as to the answer
makes reasoned guess as to strategy in part b)
uses inductive reasoning to count outcomes (say the
winning outcome was 1-2-3-4) and identifying the pattern
(P(4,4))
extends tree diagram model to solve both options for part
a)
notices similarities between formulas for permutations and
combinations
applies previous knowledge of fundamental counting
principle, permutation and combination formulas
connects permutation and combination formulas to tree
diagrams
Simplifies factorial notation before solving
Uses graphing calculator to check manual calculations
(nCr, nPr)
uses graphing calculator to perform trial and error
calculations in part b)
uses graphing calculator to solve algebraically in part b)
represents the problem using a tree diagram
solves problems using mathematical formulas for
permutations and combinations
responds to written instruction by calculating probabilities
in part a), and making adjustments to variables in part b)
presents thinking in a logical and organized manner
uses mathematical terminology correctly (arrangements,
outcomes etc.)
Communicates by combining words and diagrams
Correctly uses symbolic language of mathematics (notation
and form) - !, (), p(), n() etc.