Test 2 Study Guide
Ma 320-CRN 22189
Spring 2106
Carter
Summary First, please be aware that any study guide is only a rough outline of that which
the instructor intends to ask. Second, no matter how difficult you find these questions, be
able to do them. They are within the realm of the discussions that we’ve been having in
class so far.
1. Give a list of all the subsets of {1, 2, 3}.
2. How can this set be arranged naturally into a cube?
3. Give a list of all the divisors of 30.
4. How does this set naturally arrange naturally into the shape of a cube?
5. What is the correspondence between the divisors of 30 and the subsets of {1, 2, 3}.
6. Give a list of all the subsets of {1, 2, 3, 4} that do not include 4 as an element.
7. Give a list of all the subsets of {1, 2, 3, 4} that do include 4 as an element.
8. Give an inductive proof that the number of subsets of an n-element set (e.g. {1, 2, . . . , n})
is 2n .
9. Recall that n! is defined recursively as follows: First, 0! = 1, and subsequent values
are given by n! = n(n − 1)!. Given that n! counts the number of ways of arranging the
set {1, 2, . . . , n}, give an argument that 0! = 1 is the correct value.
10. Give an inductive proof that the number of ways of arranging the set {1, 2, . . . , n} is
n!.
11. Give the combinatorial argument that nk satisfies the recursion
n
n−1
n−1
=
+
.
k
k−1
k
(Recall that nk denotes the number of k elements subsets of an n element set).
12. Demonstrate that
n
n!
=
k
k!(n − k)!
13. Give an algebraic proof that
n!
(n − 1)!
(n − 1)!
=
+
k!(n − k)!
(k − 1)!(n − 1 − (k − 1))! k!(n − 1 − k)!
1
14. Prove that
n
2 =
n X
n
k=0
2
k