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A midterm exam for the macm 201 course at simon fraser university, department of mathematics, held in spring 2008. The exam covers various mathematical concepts, including set theory, combinatorics, and recurrence relations. Students are required to answer questions related to the principle of inclusion and exclusion, arranging letters in a word, solving equations with integer solutions, and arranging cars on a parking lot.
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MACM 201 Spring 2008 Instructor: Robert ˇS´amal February 6, 2008, 12:30 – 13:
Name: (please print) family name given name
SFU ID: student number SFU-email
Signature:
Instructions:
Question Maximum Score
[1] (a) State the Principle of Inclusion and Exclusion, and explain the used notation. Don’t use ‘... ’, nor ‘
[2] (b) Decide whether the following formulas are true or false, and if false, correct them!
[2] (c) Which elements of S are counted by N − N( c 1 c 2 c 3 ) − N( c 1 c 2 c 3 )?
word, if we require that there is
[7] (a) no consecutive fivetuple of the same letter?
[3] (b) exactly one consecutive fivetuple of the same letter?
x 1 + x 2 + x 3 + x 4 = 10
such that x 1 ,... , x 4 are integers and
[3] (a) x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 , x 4 ≥ 0?
[7] (b) 0 ≤ x 1 ≤ 4 , 0 ≤ x 2 ≤ 4 , 0 ≤ x 3 ≤ 4 , 0 ≤ x 4 ≤ 4?
an+2 + 2an+1 − 8 an = 0 (n ≥ 0), a 0 = 0, a 1 = 1.
[2] (b) Check the answer for n = 2.