1. Discrete Math for CS CMPSC 360 LECTURE 27 Last time: Counting.
Combinatorial proofs.
Today:
More counting.
Inclusion-Exclusion Principle.
CMPSC
360
11/12/2018

2. I-clicker question (frequency: BC)
In a standard deck of 52 cards, how many different ``full house'' poker hands are there? (A full house consists of five cards with three of them having one value and two having a differnt value; the order of cards doesn't matter, so {5♥,5♦, 5♠, Q♥, Q♣} and {5♦,5♥,Q♥, 5♠,  Q♣} are the same full house hand.)
13⋅6⋅6⋅4
13⋅12⋅6⋅4
13 2 ⋅11⋅ 6 2 ⋅2
52 5
None of the above
11/12/2018

3. Inclusion-Exclusion Principle
For any two sets A and B,
𝐴∪𝐵 = 𝐴 + 𝐵 − 𝐴∩𝐵 .
Example. A 3-card hand is dealt off of a standard 52-card deck. How many different such hands are there for which all 3 cards are red or all three cards are face cards?
(A face card is J,K or Q of any suit.)
11/12/2018

4. Inclusion-Exclusion Principle
For any two sets A and B,
𝐴∪𝐵 = 𝐴 + 𝐵 − 𝐴∩𝐵 .
For any three sets A ,B, and C,
𝐴∪𝐵∪𝐶 = 𝐴 + 𝐵 + 𝐶
− 𝐴∩𝐵 − 𝐵∩𝐶 − 𝐴∩𝐶
+ 𝐴∩𝐵∩C .
Example. How many integers in 1,2,…,100 are divisible by 2, 3 or 5?
=103-32=3=68
11/12/2018

5. Inclusion-Exclusion Principle
For any three sets 𝐴 1 , 𝐴 2 ,…, 𝐴 𝑛 ,
𝐴 1 ∪ 𝐴 2 ∪…∪ 𝐴 𝑛 = 𝐴 1 |+ 𝐴 2 +…+| 𝐴 𝑛
− 𝐴 1 ∩ 𝐴 2 − 𝐴 2 ∩ 𝐴 3 −…− 𝐴 𝑛−1 ∩ 𝐴 𝑛
+ 𝐴 1 ∩ 𝐴 2 ∩ 𝐴 𝐴 1 ∩𝐴 2 ∩ 𝐴 4 +…+ 𝐴 𝑛−2 ∩ 𝐴 𝑛−1 ∩ 𝐴 𝑛
−…+………
±| 𝐴 1 ∩ 𝐴 2 ∩⋯∩ 𝐴 𝑛 |
=103-32=3=68
11/12/2018