1. Inclusion-Exclusion)
Let S = {a,b,c,d} and T = {e,f,g}.
a. How many different relations on S × S are there?
b. How many different antireflexive relations on S × S are there?
c. How many different functions f : S ⟶ T are there?
d. How many different onto functions f : S ⟶ T are there?
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2. (Combinations)
A management panel at a hospital needs to include at least one member from each of the following three professions: a doctor, a lawyer and an accountant. How many different panels can be formed in each of the following situations?
a. Each profession offers 5 possible candidates. The panel size is 3.
b. Each profession offers 4 possible candidates, but A. Brent (doctor) refuses to serve with C. David (lawyer). The panel size is 3.
c. Each profession offers 5 possible candidates. The panel size is 5.
d. Each profession offers 4 possible candidates, but A. Brent (doctor) refuses to serve with C. David (lawyer). The panel size is 5.
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3. (Sequences)
How many sequences of 10 coin flips have at most 3 heads?
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4. (Advanced Counting)
How many 5-letter words over the alphabet Σ = {a, c, e, n, s}
a. include the substring ace?
b. include all letters from Σ with a before e (for example, canes)?
c. have all their letters in alphabetical order (for example, aceen)? [show answer]
5. Challenge Exercise
In a movie theatre, 4 couples are sitting in one row with men and women alternating. If no couple is sitting together, how many arrangements are possible?
