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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
- Recall the Fibonacci numbers fn = fn− 1 +fn− 2 for n ≥ 2 with f 0 = 0 and f 1 = 1. Prove that
f 0 + f 1 + · · · + fn = fn+ 2 − 1.
- Find the number of integral solutions to
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.
- Prove the Euler-Gauss identity from class:
∏^ ∞
k= 1
1 − xk^
j≥ 1
xj 2
(
∏j t= 1 1 −^ x t) 2
- Show that any collection S of 8 positive integers whose sum is 20 has a subset summing to 4 , by arguing using the pigeonhole principle that the collection must either have
(a) four 1 ’s; or (b) two 2 ’s; or (c) two 1 ’s and a 2 ; or (d) a 1 and a 3.
- Find the general solution of the recurrence relation
hn − 4hn− 1 + 4hn− 2 = 0, n ≥ 2.
- Prove the multinomial theorem: for natural numbers n we have
(x 1 + x 2 + · · · + xt)n^ =
n 1 +n 2 +···+nt =n
n n 1 n 2... nt
xn 11 · · · xn t t.
- Count the number of permutations i 1 i 2 i 3 i 4 i 5 i 6 of {1, 2, 3, 4, 5, 6} where i 1 6 = 1, 5; i 3 6 = 2, 3, 5; and i 6 6 = 5, 6.
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