CMSC 203 - Fall 2002 - Exam 2
1. Circle T if the corresponding statement is True or F if it is False.
TFThere are C(16,5) 16-bit binary words with five 1’s and eleven 0’s.
TFIf A, B, C, and D are pairwise disjoint sets, |A ∪ B ∪ C ∪ D| = |A| + |B |+ |C| + |D|.
TFThe First and Second Principles of Mathematical Induction are logically equivalent.
TFEvery recursively defined sequence must take on successively larger values.
TFIf hard-working students pass CMSC203 and Paul passed CMSC203, then we can
conclude logically that Paul is hard-working.
TFThe following is a valid argument:
¬p ∧ q
¬r → s
(r ∧ t) → ¬q
t ∨ p
--------------
∴ s
TFAllowing the top to be the 0th row, the sum of elements of the 201st row of Pascal’s
Triangle is 2201.
TFThere are 11! distinct orderings of the letters of the word MATHEMATICS.
TFFor every recursive algorithm, there is an equivalent iterative algorithm.
2. Circle V for valid or I for invalid with respect to the following arguments:
VI All dogs run fast and Zeke runs fast, therefore Zeke is a dog.
VI All dogs run fast and Zeke runs slow, therefore Zeke is not a dog.
VI All dogs run fast and Zeke is not a dog, therefore Zeke runs fast.
VI All dogs run fast and Zeke is a dog, therefore Zeke runs fast.
3. On a class field trip, a class of 10 boys and 12 girls must form a single-file line.
(a) How many distinct ways can they form this line? (b) How many distinct ways can they form this line
of all the kids of the same gender must stay grouped together? (c) How many ways can they form this
line if a certain pair of boys cannot be next to one another?
4. Suppose I have a collection of 200 pennies. (a) How many distinct ways can I create 10 piles of these
pennies? (b) How many distinct ways can I create 10 piles if each pile must have at least 5?
5. Prove one of the two Theorems below using Mathematical Induction.
Theorem 1: For all integers n > 0 and a ≠ 0,1, .
Theorem 2: Assuming Pascal’s Identity, then the sum of the entries of Row n of Pascal’s Triangle is 2n.
6. Use the Methods of Valid Arguments to obtain the indicated conclusion.
Premises: Paul does his homework or Paul watches TV.
If Paul does not watch TV, then Paul gets good grades.
If Paul gets good grades and Paul gets a good job, then Paul does his homework.
Paul does not watch TV and Paul makes the honor roll.
Conclusion: Therefore, Paul gets a good job.
7. Prove one of the two Theorems below by either Contradiction or Contraposition.
Theorem 1: There is no greatest integer.
Theorem 2: For all integers n, if n2 is odd, then n is odd.
ai
i0=
n
∑an1+ 1–
a1–
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