Test 2
Ma 320-CRN 22189
Spring 2106
Carter
Instructions. Work as many of these problems as you can and for which you have time.
Make sure you work # 11, and #14.
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 105.
4. How does this set naturally arrange naturally into the shape of a cube?
5. What is the correspondence between the divisors of 105 and the subsets of {1, 2, 3}.
6. Give a list of all the subsets of {1, 2, 3} that do not include 3 as an element.
7. Give a list of all the subsets of {1, 2, 3} that do include 3 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)!
14. Prove that
n
2 =
n X
n
k=0
1
k