1. Counting by Complement and the Inclusion/Exclusion Principle
Sandy Irani
ICS 6D

2. 5-card Hands
How many 5-card hands have exactly 1 club?

3. 5-card Hands
How many 5-card hands have at least one club?

4. Counting by Complement
Set S of items.
Let P ⊆ S be the set of items in S that have some particular propery:
|S| - |P| = |P|
Set of all 5-card
hands with at
least one club
Set of all 5-card
hands
Set of all 5-card
hands with
no clubs

5. Counting by Complement: Examples
How many length 8 strings over the alphabet {a, b, c} have at least one “a”?

6. Counting by Complement: Examples
A software team has 10 senior member and 10 junior members. Must select a set of 4 people to work on a project. How many selections have at least one junior member?

7. - More Donut Selection =
How many ways to select 20 donuts from 4 varieties. There is a large selection of glazed, jelly, and maple. But there are only 5 chocolates left. (# chocolates must be ≤ 5)
Number of selections
with at more than 5
chocolate donuts
Number of selections
with at most 5
chocolate donuts
Number of selections
with no restrictions - =

8. Solution to Sums of Variables
How many solutions are there to the following equation, where each variable xi is a non-negative integer?
x1 + x2 + x3 + x4 = 12
x2 ≤ 3

9. Solution to Sums of Variables
How many solutions are there to the following equation, where each variable xi is a non-negative integer?
x1 + x2 + x3 + x4 = 12
x2 ≤ 3 and x4 ≥ 2

10. The Sum Rule (Review) For finite sets A1, A2,…, An ,
If the sets are pairwise disjoint (Ai ∩ Aj = φ, for i≠j)
then |A1 ∪ A2 ∪ … ∪ An|= |A1| + |A2| + … + |An|
What if the sets are not pairwise disjoint?

11. Inclusion/Exclusion 2 Sets
|A ∪ B| = |A| + |B| - |A ∩ B|
S general population of elements
P1 is the set of elements with property 1
P2 is the set of elements with property 2
How many elements in S have property 1 or 2 (inclusive or)?
| P1 ∪ P2| = Number of elements with property 1
+ Number of elements with property 2
- Number of elements with both properties.

12. Inclusion/Exclusion Example
How many 5-card hands from a standard playing hand have exactly one King or exactly one Ace (or both)? ∪

13. Inclusion/Exclusion Example
How many strings of length 6 over the alphabet {A, B, C} start with a C or end with a C? (inclusive or)

14. Inclusion/Exclusion Example
How many strings of length 6 over the alphabet {A, B, C} start with a B or C? (inclusive or)

15. Inclusion/Exclusion Example
How many strings of length 6 over the alphabet {A, B, C} have at least 5 consecutive A’s?

16. Inclusion/Exclusion with 3 Sets
|A ∪ B ∪ C| = |A| + |B| + |C|
- |A ∩ B| - |A ∩ C| - |B ∩ C|
+ |A ∩ B ∩ C|

17. Inclusion/Exclusion with 3 Sets
Drug test on a population of 1000 people
122 people develop symptom A
88 people develop symptom B
112 people develop symptom C
27 people develop symptom A and B
29 people develop symptom A and C
32 people develop symptom B and C
10 people develop all three symptoms
How many people get at least one symptom?

18. Inclusion/Exclusion with 3 Sets
Line up of 7 people:
Mother, Father, 3 sons, 2 daughters
How many line-ups are there in which the mother is next to at least one of her 3 sons?

19. Inclusion/Exclusion Example
How many strings of length 6 over the alphabet {A, B, C} have at least 4 consecutive A’s?

20. Incl/Excl 3 Sets
How many integers in the range 1 through 42 are divisible by 2, 3, or 7?

21. Inclusion/Exclusion with 4 Sets
|A ∪ B ∪ C ∪ D | = |A| + |B| + |C| + |D|
- |A ∩ B| - |A ∩ C| - |B ∩ C|
- |A ∩ D| - |B ∩ D| - |C ∩ D|
+ |A ∩ B ∩ C| + |A ∩ B ∩ D|
+ |A ∩ C ∩ D| + |B ∩ C ∩ D|
- |A ∩ B ∩ C ∩ D|

22. Inclusion/Exclusion with 4 Sets
Suppose you are using the inclusion-exclusion principle to compute the number of elements in the union of four sets.
Each set has 15 elements.
The pair-wise intersections have 5 elements each.
The three-way intersections have 2 elements each.
There is only one element in the intersection of all four sets. What is the size of the union?
What is the size of the union?

23. Incl/Excl and counting by complement
How many 5-card hands have at least one ace or at least one queen (inclusive or)?