COUNTING: SUMMARY
Tree Diagrams often serve as a useful
visual aid for many counting problems
This tree diagram shows that if we toss a penny, nickel and dime,
there are 8 possible outcomes:
{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Visual aid for ‘2 dice’ problems
Although a tree diagram will work for
problems involving rolling 2 dice, the grid
below is often the best visual aid for this
type of problem.
11
21
31
41
51
61
12
22
32
42
52
62
13
23
33
43
53
63
14
24
34
44
54
64
15
25
35
45
55
65
16
26
36
46
56
66
OTHER COUNTING TOOLS:
1. THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first sub-task can be performed in a ways, the second
one in b ways, the third in c ways, and so on, can be done in a x b x c x ... ways.
2. PERMUTATIONS
A permutation is an ordering of distinct objects (usually pictured as being in a straight line).
If we select r different objects from a set of n objects and arrange them in a particular order, we call this a
permutation of n objects taken r at a time and is denoted P(n,r).
A permutation for which r = n, P(n, n), is called a factorial.
FACTORIAL [ n! = P(n, n) ]
n! = n x (n-1) x (n-2) x ... x 1
(and 0! = 1)
3. COMBINATION
A combination is a subset of a set of distinct objects (where order is not important). If we select r objects from
a set of n objects, we call this a combination of n objects taken r at a time and is denoted C(n,r).
Sometimes a problem requires the use of a combination of counting tools.
The computing committee at a college has of 5 administrators, 7 faculty and 5 students. A four-person
subcommittee is to be formed. The chair and vice chair of the subcommittee must be committee administrators
but the rest of the subcommittee can be composed of any of the faculty or students on the main committee. In
how many ways can this subcommittee be formed?
Solution: For part, order is important (selecting the chair & vice chair) & for the rest of the subcommittee, order
is not important. So, we need to use the permutation formula to find how many ways there are to select the
chair and vice chair and then use the combination formula to find how many ways there are to select the rest of
the subcommittee. We will use the FCP to compute the total number of ways to do both together.
By the FCP:
P( 5, 2)
x
C(12, 2)