STAT101 Worksheet: Combinations, Permutations, etc.
4
KEY
Factorials
Calculate 8!
8! = 8*7*6*5*4*3*2*1 = 40,320
Combinations & Permutations
n
Pr 
n!
(n  r )!
n
Cr 
n!
(n  r )! r!
Mountain Bicycles: A bicycle shop owner has 12 mountain bicycles in the showroom. The owner wants to show
five of them at a bicycle trade show. How many different ways can a group of five be selected?
12
C5 
12!
12 *11 *10 * 9 * 8 * 7! 11 * 9 * 8


 792
(12  5)!*5!
7!*5 * 4 * 3 * 2 *1
1
Mountain Bicycles: A bicycle shop owner has 12 mountain bicycles in the showroom. How many different ways
can he place five of these bicycles in his shop display window?
P 
12 5
12!
12 *11 *10 * 9 * 8 * 7! 12 *11 *10 * 9 * 8


 95040
(12  5)!
7!
1
Child #
Tree Diagrams:
A Tree Diagram systematically lists all possible ways a sequence of events can occur.
Family: What is the probability that at least one boy will be born into a family having three
children?
1
2
3
B
B
G
B
B
G
Probability of at least one boy = 7/8 = .875 or 87.55
G
B
B
G
G
B
G
G
Multiplication Rule for Independent Events:
In a sequence of n events in which the first event has k1 possibilities, the second event k2, etc. (to kn), the total number
of possibilities is k1* k2* …* kn-1* kn.
Types of Paint: A manufacturer wishes to manufacture several different paints. Considerations in making the paint are: 1)
Color (red, blue, green, black, white, brown, yellow); 2) Type (latex, oil); 3) Texture (flat, semigloss, high gloss); and 4) Use
(interior, exterior). How many different kinds of paint can be made? [source: Bluman, 4ed ,p. 219]
k1 = color = 7
k2 = Type = 2
k3 = Texture = 3
k1 * k2 * k3 * k4 = 7 * 2 * 3 * 2 = 84 paint types
k4 = Use = 2
Practice Problems
Factorials:
Calculate 5!
5*4*3*2*1 = 120
Calculate 0!
0! = 1
Combinations & Permutations:
Dress Design: A dress designer wants to combine two fabric colors from among six in the creation of a new dress. How
many different possibilities are there in this scenario?
6
C2 
6!
6 * 5 * 4! 3 * 5


 15
(6  2)!*2! 4!*2 *1
1
Awards: Suppose that in your lifetime you win a Pulitzer, Oscar and a Grammy. You decide to place these awards on your
fireplace mantel. How many ways could you do this?
P 
3 3
3!
3 * 2 *1

6
(3  3)!
1
Random Samples: A class has 30 students. Determine the number of samples of size ten that may be obtained.
30
C10 
30!
30 * 29 * 28 * 27 * 26 * 25 * 24 * 23 * 22 * 21* 20! 29 * 7 * 3 *13 * 5 * 23 *11* 3


 30,045,015
(30  10)!*10!
20!*10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 *1
1
Horse Racing: In horse racing, one can make a trifecta bet by specifying which horse will come in first, second, and third, in
the correct order. One can make a box trifecta bet by specifying which three horses will come in first, second, and third
without specifying the order. [Navidi, 2013, p. 232]
a)
In an eight-horse field, how many different ways can one make a trifecta bet?
P 
8 3
8!
8 * 7 * 6 * 5!

 336
(8  3)!
5!
b) In an eight-horse field, how many different ways can one make a box trifecta bet?
8
C3 
8!
8 * 7 * 6 * 5! 8 * 7


 56
(8  3)!*3! 5!*3 * 2 *1
1
Ice Cream: A certain ice cream parlor offers 15 flavors of ice cream. You want an ice cream cone with three scoops of ice
cream, all different flavors. [Navidi, 2013, p. 232]
a)
In how many ways can one order an ice cream cone if it matters which flavor is on top, in the middle, and on
the bottom?
P 
15 3
15!
15 *14 *13 *12!

 2730
(15  3)!
12!
b) In how many ways can you chose a cone if the order of the favors does not matter?
15
C3 
15!
15 *14 *13 *12! 5 * 7 *13


 455
(15  3)!*3!
12!*3 * 2 *1
1
Tree Diagrams:
Coins and Dice: Draw a tree diagram that presents the flipping of a coin combined with the roll of
one die.
Dinner: Draw a tree diagram for a meal consisting of four appetizers, five entrees, and five drinks (too
full for dessert) and see why this type presentation is replaced by the multiplication rule. No tree
drawn. 4*5*5 = a tree with 100 meal combinations.
H
T
1
2
3
4
5
6
1
2
3
4
5
6
Multiplication Rule (indep. events):
Blood Types: There are four blood types (A, B, AB, O) and two Rh factors (Rh+, Rh-). If donors are further identified by
gender (male female), how many different ways can a donor have his or her blood classified? 4*2*2 = 16 classifications
Panera’s: You decide to go to Panera’s for lunch to have a pick-two meal plus a drink. There are seven beverages to choose
from. For the pick-two meal there are six soups and eight sandwiches from which to choose. From how many different meal
combinations could you choose? 7*6*8 = 336 meal combinations
Genes: Human genetic material (DNA) is made up of a sequence of the molecules adenosine (A), guanine (G), cytosine (C),
and thymine (T), which are called bases. A codon is a sequence of three bases. Replicates are allowed, so AAA, CGC, etc.
are codons. Codons are important because each codon causes a different protein to be created. [Navidi, 2013, p.233]
a) How many different codons are there? 4*4*4 = 64
b) How many different codons are there in which all three codons are different? 4*3*2 = 24
c) The bases A and G are called purines, while C and T are called pyrimidines. How many different codons are
there in which the first base is a purine and the other two are pyrimidines? (Try a tree diagram here.)
2*2*2 = 8
License Plates: In New York most automobile license plates have four numbers (0 thru 9) and three letters (A thru Z).
a)
How many different license plate numbers can New York State create?
10*10*10*10*26*26*26* = 175,760,000 plates
b) My car’s letters are ELN. How many other cars could have ELN as part of its plate number?
ELN = one alpha result. 1*10*10*10*10 = 10,000 plates with ELN as the alpha portion