12.2 COMBINATIONS
Question 1: How many arrangements of 4 people in 3 chairs can we make, where order doesn’t
matter?
Method 1: List all Possiblilities
ALL THE PERMUTATIONS ARE:
A
A
A
A
A
A
B
B
C
C
D
D
C
D
B
D
C
B
C
C
C
C
C
C
A
A
B
B
D
D
B
D
A
D
A
B
B
B
B
B
B
B
A
A
C
C
D
D
C
D
A
D
A
C
D
D
D
D
D
D
A
A
B
B
C
C
B
C
A
C
A
B
CIRCLE ALL THE COMBINATIONS, DISREGARDING ALL THE ARRANGEMENTS
THAT MAY BE THE SAME BUT IN A DIFFERENT ORDER
Number of Distinct Combinations: __________
Number of times each combination occurred in our chart ______
You try (Note: The order doesn’t matter)
5 people in 3 chairs
7 people in 2 chairs
12.2 COMBINATIONS
Practice
1.) How many different 5 card hands are there in poker (1 deck)?
2.) How many different 3 card hands can be dealt from a deck of 52 cards?
3.) An English teacher must select six of fifteen books for her students to read
in the first quarter. How many groups of six books can be selected ?
4.) A co-ed basketball team has 4 men and 3 women. If exactly 2 women must
always be on the court, how many different groups of players can be out on the
court ?
5.) From a group of 10 men and 12 women, how many committees of 5 men and 6 women
can be formed ?
6.) In a standard deck of 52 playing cards, in how many 5 card hands are all 5 cards the
same suit (a “flush”) ?
7.) A restaurant serves omelets that can be ordered with any of the ingredients below:
Omelets $3.00
(plus $.50 for each ingredient)
Vegetarian
green pepper
red pepper
onion
mushroom
tomato
cheese
Meat
ham
bacon
sausage
steak
a.) Suppose you want exactly two vegetarian ingredients and one meat ingredient.
How many different types of omelets can you order ?
b.) Suppose you can afford at most three ingredients on your omelet. How many
Different types of omelets can you order ?
12.2 COMBINATIONS
Binomial Expansion Techniques
Equivalent Triangle
Binomial Theorem
Pascal’s Triangle
=
Expand each of the following (watch for a pattern)
( a + b) 2 =
( a + b) 3 =
( a + b) 4 =
Use Pascal’s triangle to expand ( x + y)5
Now use the pattern that you have identified to expand the following.
( x + 5) 4
(2 x + 3) 3
(2 x − 1) 5
12.2 COMBINATIONS