Section 10.2A
Combinations
52!

52 C 5 
47! 5!
2,598,960
How are combinations different from permutations?
From a group of 5 students (Kyle, Joanne, Mary,
Charlie, and Vicki) we need to select 3…
…who will get a first, second, and third place
award for their work in math.
…who will help their teacher to arrange the
desks.
Is order relevant to the arrangements?
Combinations
of r objects, from a group of n items
(order is NOT relevant to the problem)
n
C
r
n!
C

n r
(n  r )!r !
Permutations
of r objects, from a group of n items
(order IS relevant to the problem)
n
P
r
n!
P

n r
( n  r )!
How are combinations different from permutations?
From a group of 5 students (Kyle, Joanne, Mary,
Charlie, and Vicki) we need to select 3…
…who will get a first, second, and third place
award for their work in math.
5!
 60
5 P3 
2!
…who will help their teacher to arrange the
desks.
5!
5
C3 
2!3!
 10
Is order relevant to the arrangements?
MULTIPLE EVENTS: when finding the number of ways
both an event A and an event B can occur, you need to
multiply. When finding the number of ways that event A
or event B can occur, you add instead.
A restaurant serves omelets that can be ordered
with any of the ingredients shown.
a) Suppose you want to order exactly 2
vegetarian ingredients and 1 meat ingredient in
your omelet. How many different types of
omelets can you order?
a) 2 vegetarians and 1 meat… 6 C 2  4 C1  60
MULTIPLE EVENTS: when finding the number of ways
both an event A and an event B can occur, you need to
multiply. When finding the number of ways that event A
or event B can occur, you add instead.
A restaurant serves omelets that can be ordered
with any of the ingredients shown.
a) Suppose you want to order exactly 2
vegetarian ingredients and 1 meat ingredient in
your omelet. How many different types of
omelets can you order?
b) Suppose you want to order 2 vegetarian
ingredients or 3 meat ingredients in your
omelet. How many different types of omelets
can you order?
b) 2 vegetarian or 3 meat:
6
C2  4 C3  15  4  19
MULTIPLE EVENTS: when finding the number of ways
both an event A and an event B can occur, you need to
multiply. When finding the number of ways that event A
or event B can occur, you add instead.
A restaurant serves omelets that can be ordered
with any of the ingredients shown.
a) Suppose you want to order exactly 2
vegetarian ingredients and 1 meat ingredient in
your omelet. How many different types of
omelets can you order?
b) Suppose you want to order 2 vegetarian
ingredients or 3 meat ingredients in your
omelet. How many different types of omelets
can you order?
c) Suppose you want to order at most 2
ingredients in your omelet. How many different
types of omelets can you order?
c) 0 ingredients or 1 or 2
10
C0 10 C1 10 C2  1  10  45  56
You are going on a vacation. You are packing your
clothes and you can take as many as 7 t-shirts
and 5 pairs of shorts. If you want to take exactly 3
pairs of shorts and 4 t-shirts, how many different
ways can you pack?
5!
7!

 350
5 C 3  7 C4 
2! 3! 3! 4!
Your school football team has 11 scheduled
games for the season. You want to attend at least
9 games. How many different combinations of
games can you attend?
Reword the problem:
attend 9 games or 10 games or 11 games.
11
C9 11C10 11C11  55  11  1  67
Additional examples:
Find the number of possible 5-card hands that contain
the cards specified:
a) 2 aces and 3 kings
4
C 2 4 C 3  24
b) 5 hearts or 5 diamonds
13
C5 13 C5  2,574
c) 3 hearts (hint: this means 3 hearts and 2 “non-hearts”)
13
C 3 39 C 2  211,926
d) 4 face cards (hint: this means 4 face cards and 1 “non-face” card)
12
C 4  40 C1  495  40  19,800