1. Discrete Math for CS CMPSC 360 LECTURE 27 Last time: Counting. Today:
Combinatorial proofs.
CMPSC
360
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2. 1st rule of counting
If an object can be formed by a succession of 𝑘 choices,
where there are 𝑛 1 ways of making the first choice, and
for every way of making the 1st choice, there are 𝑛 2 ways of making the 2nd choice, and
for every way of making the 1st and 2nd choices there are 𝑛 3 ways of making the 3rd choice, and
so on up to the 𝑛 𝑘 -th choice, then
the number of distinct objects that can be made in this way is
𝑛 1 ⋅ 𝑛 2 ⋅ 𝑛 3 ⋅…⋅ 𝑛 𝑘 .
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3. 2nd rule of counting
Suppose an object can be formed by a succession of choices, and the order in which the choices are made does not matter.
Let A be the set of possible objects if the order of choices matters.
Let B the set of possible objects if it does not.
Suppose we can map all objects from A to objects in B is that exactly 𝑡 elements of A are mapped to every element of B.
Then |𝐵|= 𝐴 𝑡 .
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4. Typical counting problems
Order 𝑛 elements: 𝑛! ways.
Choose 𝑘 out of 𝑛 without replacement, ordered:
𝑛! 𝑛−𝑘 ! ways.
Choose 𝑘 out of 𝑛 without replacement, unordered:
𝑛 𝑘 = 𝑛! 𝑘! 𝑛−𝑘 ! ways.
Choose 𝑘 out of 𝑛 with replacement, ordered: 𝑛 𝑘 ways.
Choose 𝑘 out of 𝑛 with replacement, unordered: ???
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5. I-clicker question (frequency: BC)
How many undirected graphs with vertex sets 1,2,…,𝑛 are there?
𝑛 2
2 𝑛
2 𝑛 𝑛−1 /2
2 𝑛 𝑛−1
2 𝑛 2
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6. I-clicker question (frequency: BC)
How many directed graphs with vertex sets 1,2,…,𝑛 are there?
𝑛 2
2 𝑛
2 𝑛 𝑛−1 /2
2 𝑛 𝑛−1
2 𝑛 2
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7. Typical counting problems
Order 𝑛 elements: 𝑛! ways.
Choose 𝑘 out of 𝑛 without replacement, ordered:
𝑛! 𝑛−𝑘 ! ways.
Choose 𝑘 out of 𝑛 without replacement, unordered:
𝑛 𝑘 = 𝑛! 𝑘! 𝑛−𝑘 ! ways.
Choose 𝑘 out of 𝑛 with replacement, ordered: 𝑛 𝑘 ways.
Choose 𝑘 out of 𝑛 with replacement, unordered:
𝑛+𝑘−1 𝑘 ways.
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8. A combinatorial proof To prove an equality of the form LHS=RHS:
Find a counting question you can answer in two ways.
Argue that the first way gives LHS.
Argue that the second way gives RHS.
Conclude that LHS=RHS.
It is like playing Jeopardy!.
Ex.1. 𝑛 𝑘+1 = 𝑛−1 𝑘 + 𝑛−2 𝑘 +…+ 𝑘 𝑘
Ex.2. 𝑛 𝑘 = 𝑛−1 𝑘−1 + 𝑛−1 𝑘
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