







Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Material Type: Exam; Professor: Yong; Class: Intro to Combinatorics; Subject: Mathematics; University: University of Illinois - Urbana-Champaign; Term: Fall 2009;
Typology: Exams
1 / 13
This page cannot be seen from the preview
Don't miss anything!
U. of Illinois MATH 413 Final Exam – Fall 2009
Answer as many problems as you can. Each question is worth 6 points (total points is 66 ). Show your work. An answer with no explanation will receive no credit. Write your name on the top right corner of each page. No external assistance permitted, including calculators of any kind. [Total time: 3 hours]
Below are some formulas you might find helpful. This list is neither guaranteed to be complete, nor do I guarantee that each formula is actually needed on the exam.
( n k
n − 1 k
n − 1 k − 1
j≥ 0
xj^ =
1 − x
∑^ n
j= 0
xj^ =
1 − xn+^1 1 − x
(x + y)n^ =
∑^ n
j= 0
n k
xjyn−j, n ∈ N
( 1 − x)−n^ =
k≥ 0
n + k − 1 n − 1
xk, n ∈ N
|A 1 ∪A 2 ∪· · ·∪An| =
i 1
|Ai 1 |−
i 1 ,i 2
|Ai 1 ∩Ai 2 |+· · ·+(− 1 )n^
i 1 ,...,in
|Ai 1 ∩· · ·∩Ain |
|Ac 1 ∩Ac 2 ∩· · ·∩Acn| = |S|−
i 1
|Ai 1 |+
i 1 ,i 2
|Ai 1 ∩Ai 2 |+· · ·+(− 1 )n+^1
i 1 ,...,in
|Ai 1 ∩· · ·∩Ain |
1
f 0 + f 1 + · · · + fn = fn+ 2 − 1.
x 1 + x 2 + x 3 + x 4 = 18
subject to 1 ≤ x 1 ≤ 5 , − 2 ≤ x 2 ≤ 4 , 0 ≤ x 3 ≤ 5 and 3 ≤ x 4 ≤ 9.
∏^ ∞
k= 1
1 − xk^
j≥ 1
xj 2
(
∏j t= 1 1 −^ x t) 2
(a) four 1 ’s; or (b) two 2 ’s; or (c) two 1 ’s and a 2 ; or (d) a 1 and a 3.
hn − 4hn− 1 + 4hn− 2 = 0, n ≥ 2.
(x 1 + x 2 + · · · + xt)n^ =
n 1 +n 2 +···+nt =n
n n 1 n 2... nt
xn 11 · · · xn t t.
(This page has been intentionally left blank)