Description
problem 1 (Combinations and Permutations). (a) There are four different bike routes between Brookline and Cambridge. In how many distinct ways can you make a round-trip from Brookline to
Cambridge and back? In how many ways can you make this trip if you take a different route on the
way back?
(b) You want to arrange 4 different flower vases in a row on your shelf. How many distinct arrangements
can you come up with?
(c) You have meal prepped three types of main dishes and four types of appetizers. How many distinct
meals can you have during the week, if a meal = main dish + appetizer.
Solution: Your solution here.
problem 2 (Counting). A standard deck of cards has 52 cards. Each card has a rank and a suit. There
are 13 ranks: A (Ace), 2, 3, 4, 5, 6, 7, 8, 9, 10, J (Jack), Q (Queen), K (King). There are 4 suits: clubs (♣),
diamonds (♦), hearts (♥), and spades (♠).
We draw 3 cards from a standard deck without replacement. How many possibilities are there if:
(a) the cards have the same rank;
(b) the cards have different ranks;
(c) two of the cards have the same rank and the third has a different rank.
Solution: Your solution here.
problem 3 (Induction). Show by induction on n that
1 + r + r
2 + · · · + r
n =
r
n+1 − 1
r − 1
,
for all n ∈ N and r 6= 1. (N denotes the set of all natural numbers. In this class, we adopt the convention
that N includes 0.)
Solution: Your solution here.
problem 4 (Proofs by contradiction). Prove by contradiction that if a
2
is even then a is even.
Solution: Your solution here.
problem 5 (Set Operations). Let A, B, C be sets. Show that
(A \ B) \ C = (A \ C) \ (B \ C) .
Solution: Your solution here.
problem 6 (Calculus). Consider the following function f : R → R:
f(x) =
x
2 − 2x − 1 for x ∈ [0, 5]
3 for x ∈ [7, 10]
0 otherwise
(a) Sketch the graph of the function.
(b) For each of the following intervals [a, b], evaluate the integral R b
a
f(x)dx: [−1, 3], [1, 8], [6, 12].
(c) Find the derivative of f wherever it exists, and clearly identify at which points the derivative does
not exist.
(d) Let g : R → R be a differentiable function and let g
0 be its derivative. Let h be the composition of g
with f , i.e., the function h : R → R such that h(x) = g(f(x)) for all x ∈ R. Find the derivative of h
wherever it exists, expressed in terms of g
0
.
Solution: Your solution here.
H1-2
