4_7Combinations.notebook
April 24, 2013
From yesterday's practice questions...
How many 4­letter "words" can be made using the letters QUIP?
How many 4­letter "words" can be made using the letters QUIP if Q and U must be together? NOTE: Always deal with restrictions first
and "blocked" objects are considered one item
1
4_7Combinations.notebook
April 24, 2013
How many different 6­letter "words" can be made from the word OTTAWA?
At first glance we might think there are 6! different "words"
But there are multiples of some letters. We are not able to distinguish between one "T" and the other "T" Since there are 2! ways to arrange the Ts and 2! ways to arrange the As, then there will be 2! x 2! duplicates.
To account for this over­counting we simply divide
6! / (2! x 2!) or 6! / (2!2!)
2
4_7Combinations.notebook
April 24, 2013
Sometimes it is more efficient to use the COMPLEMENT
(All possible arrangements) ­ (arrangements you don't want) = (arrangements you do want)
Brooklyn has 6 different coloured sweaters to pile up in her drawers. How many ways can she stack them so that the red sweater is not on the bottom?
Using Direct Reasoning
TIP Deal wit the restriction
The red sweater cannot b
sweaters that can be on the Since there are 5 other sw
we know the number of arra
Answer: 5 x 5! = Using Indirect Reasoning (
All ­ Red sweater on botto
If there were no restriction
If the Red sweater was on Red sweater NOT on the b
3
4_7Combinations.notebook
April 24, 2013
4
4_7Combinations.notebook
April 24, 2013
5
4_7Combinations.notebook
April 24, 2013
Combinations
A permutation is an arrangement of objects that need to be in a particular order. (role/job/position)
A combination is an arrangement of objects that do NOT need to be in order.
Permutation example: Choosing a President and a Vice­president from a group of 6 people solution: P(6,2) = 6! / (6­2)! = 6! / 4! = 6x5 = 30 possible arrangements
Combination example: Choosing two people for a committee from a group of 6 people
In this case if we choose Bob and Julie, it would be the same as choosing Julie and Bob (order is not important = "combination")
If Bob and Julie were chosen as President and Vice­President, you can see that President Bob and VP Julie is not the same as President Julie and VP Bob. (specific roles = "permutation")
6
4_7Combinations.notebook
April 24, 2013
7
4_7Combinations.notebook
April 24, 2013
From a class of 30 students, determine the number of ways a 5­
person committee can be selected to organize a class party.
a) with no restrictions
Hint: Choose 5 people from a group of 30, no order
b) with Marnie on the committee
HINT: Marnie must be included, so we are choosing 4 other people from the remaining 29, no order
8
4_7Combinations.notebook
April 24, 2013
In Lotto 649 you need to choose 6 numbers from a possible 49. You do not have to match them in the same order.
In Super 7 you choose 7 numbers out of 47
a) How many number combinations can be picked in Lotto 649?
Hint: 49 numbers, choose 6
b) Which game do you have a better chance of winning?
HINT: Super 7 has 47 numbers, choose 7
9
4_7Combinations.notebook
April 24, 2013
From a group of 15 elementary students, 11 high school students and 12 post­secondary students, how many ways can a committee of 5 students be chosen?
a) if there are no restrictions?
b) if the committee must be only high school students?
HINT: choose 5 from a t
HINT: choose 5 from a t
HINT: OR means to AD
c) if it must be only high school or elementary students?
HINT: choose 1 from 12
d) if the committee must have one post­secondary, two high school and two elementary students? HINT: choose 1 from 12 post secondary AND choose 4 from the remaining
e) if the committee must have exactly one post­secondary student?
f) if Michelle must be on the committee?
HINT: Michelle must be on the committee means we only choose 4 from the remaining group of 37
g) if Steven and Chadd must be on the committee?
HINT: We only choose 3 more from the remaining group of 36
h) if Steven and Chadd must NOT be on the committee together?
HINT: use the complement: All ­ (Steven and Chadd)
or use direct reasoning: (without Steven and Chadd) or (with just Steven) or (with just Chadd)
i) if there must be at least one elementary student?
HINT: at least one elementary means: could be one elementary, could be two elementary, three elementary, ... or all five elementary
or use complement: all ­ (no elementary)
10