Combinatorics
Problem: How to count without counting.
•How do you figure out how many things there are with
a certain property without actually enumerating all of
them.
Sometimes this requires a lot of cleverness and deep math-
ematical insights.
But there are some standard techniques.
•That’s what we’ll be studying.
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Sum and Product Rules
Example 1: In New Hampshire, license plates consisted
of two letters followed by 3 digits. How many possible
license plates are there?
Answer: 26 choices for the first letter, 26 for the second,
10 choices for the first number, the second number, and
the third number:
262×103= 676,000
Example 2: A traveling salesman wants to do a tour of
all 50 state capitals. How many ways can he do this?
Answer: 50 choices for the first place to visit, 49 for the
second, . . . : 50! altogether.
Chapter 4 gives general techniques for solving counting
problems like this. Two of the most important are:
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The Sum Rule: If there are n(A) ways to do Aand,
distinct from them, n(B) ways to do B, then the number
of ways to do Aor Bis n(A) + n(B).
•This rule generalizes: there are n(A) + n(B) + n(C)
ways to do Aor Bor C
•In Section 4.8, we’ll see what happens if the ways of
doing Aand Baren’t distinct.
The Product Rule: If there are n(A) ways to do A
and n(B) ways to do B, then the number of ways to do
Aand Bis n(A)×n(B). This is true if the number of
ways of doing Aand Bare independent; the number of
choices for doing Bis the same regardless of which choice
you made for A.
•Again, this generalizes. There are n(A)×n(B)×n(C)
ways to do Aand Band C
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Some Subtler Examples
Example 3: If there are nSenators on a committee, in
how many ways can a subcommittee be formed?
Two approaches:
1. Let N1be the number of subcommittees with 1 sen-
ator (n), N2the number of subcommittees with 2
senator (n(n−1)/2), . . .
According to the sum rule:
N=N1+N2+· · · +Nn
•It turns out that Nk=n!
k!(n−k)! (nchoose k); this
is discussed in Section 4.4
•A subtlety: What about N0? Do we allow sub-
committees of size 0? How about size n?
◦The problem is somewhat ambiguous.
If we allow subcommittees of size 0 and n, then
there are 2nsubcommittees altogether.
◦This is the same as the number of subsets of the
set of nSenators: there is a 1-1 correspondence
between subsets and subcommittees.
2. Simpler method: Use the product rule!
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