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Quizzes > Engineering & Technology

Intro To Combinatorics Quiz

Free Practice Quiz & Exam Preparation

Difficulty: Moderate
Questions: 15
Study OutcomesAdditional Reading
3D voxel art symbolizing Intro to Combinatorics course material

Test your combinatorial skills with this engaging practice quiz for Intro to Combinatorics. Designed for both undergraduate and graduate learners, the quiz covers essential topics such as permutations and combinations, generating functions, recurrence relations, inclusion and exclusion, Polya's theory of counting, and block designs to boost your problem-solving prowess and course confidence.

What is a permutation?
An arrangement of objects where order matters.
A selection of objects where order does not matter.
A method for counting subsets regardless of order.
A process of grouping objects into unordered sets.
What is a combination?
A selection of objects where order does not matter.
An arrangement of objects where order matters.
A process that counts the number of sequences.
A method for ordering objects into sequences.
How many distinct arrangements (permutations) does a set of n distinct items have?
n
n²
n!
2❿
What is the generating function for the sequence (1, 1, 1, 1, ...)?
1/(1 - x)
1/(1 + x)
1 - x
1 + x
Which statement best describes a recurrence relation?
It defines each term of a sequence based on its preceding terms.
It provides an explicit formula for the nth term of a sequence.
It exclusively counts the number of permutations of objects.
It is used to generate power series for combinatorial sequences.
Using the Principle of Inclusion-Exclusion, how many integers from 1 to 100 are divisible by 2 or 3?
67
68
66
64
Consider the recurrence relation a(n) = 3a(n-1) + 2 with a(0) = 1. Which generating function G(x) represents this sequence?
(1 + x)/((1 - x)(1 - 3x))
1/((1 - 3x)(1 - x))
(1 - x)/((1 + x)(1 - 3x))
(1 + 2x)/((1 - x)(1 - 3x))
Using the Principle of Inclusion-Exclusion, how many numbers between 1 and 100 are divisible by 2, 3, or 5?
74
76
72
70
Which of the following best describes Pólya's Enumeration Theorem in counting colorings?
It counts distinct colorings by considering symmetry and group actions.
It provides a method for counting restricted permutations.
It sums the total number of subsets of a set.
It solves recurrence relations by using generating functions.
What is the number of distinct arrangements of the letters in the word 'BANANA'?
60
120
180
360
In a balanced block design, which parameter represents the number of blocks in which each treatment appears?
r
b
k
λ (lambda)
What is the coefficient of x❴ in the expansion of (1 + x)❷?
35
21
28
56
Solve the recurrence relation a(n) = 2a(n-1) + 3 with a(0) = 1 and provide its closed form.
4·2❿ - 3
2❽❿❺¹❾ - 3
4·2❿ + 3
4·2❽❿❻¹❾ - 3
Which generating function represents the partition function p(n), where p(n) counts the number of partitions of n?
∝ (from k=1 to ∞) [1/(1 - xᵝ)]
∑ (from k=1 to ∞) [xᵝ/(1 - xᵝ)]
1/(1 - x)
((1 - x)/(1 - 2x))
If you need to select 3 out of 5 people to form a committee, and one specific person must always be included, how many ways can the committee be formed?
6
10
4
8
0
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Study Outcomes

  1. Apply permutation and combination techniques to solve discrete counting problems.
  2. Synthesize generating functions and recurrence relations to model and analyze combinatorial scenarios.
  3. Utilize inclusion - exclusion and Polya's counting theory to address complex counting challenges.

Intro To Combinatorics Additional Reading

Here are some top-notch academic resources to supercharge your combinatorics journey:

  1. Analytic Combinatorics by Princeton University Dive into the world of combinatorial structures and generating functions with this comprehensive course led by Professor Robert Sedgewick. Perfect for those looking to deepen their understanding of analytic methods in combinatorics. ([coursera.org](https://www.coursera.org/learn/analytic-combinatorics?utm_source=openai))
  2. Combinatorics and Probability by University of California San Diego Explore the fundamentals of counting, binomial coefficients, and probability in this engaging course. It's a great starting point for beginners and offers practical applications of combinatorial concepts. ([coursera.org](https://www.coursera.org/learn/combinatorics?utm_source=openai))
  3. Combinatorial Theory: Introduction to Graph Theory, Extremal and Enumerative Combinatorics by MIT This course provides an in-depth look at modern combinatorial topics, including graph theory and enumeration, with a focus on applications and connections to other fields. ([ocw.mit.edu](https://ocw.mit.edu/courses/18-315-combinatorial-theory-introduction-to-graph-theory-extremal-and-enumerative-combinatorics-spring-2005/?utm_source=openai))
  4. Algebraic Combinatorics Lecture Notes by MIT Access detailed lecture notes covering topics like Catalan numbers, Young tableaux, and q-binomial coefficients. These notes are a valuable resource for understanding the algebraic aspects of combinatorics. ([ocw.mit.edu](https://ocw.mit.edu/courses/18-212-algebraic-combinatorics-spring-2019/pages/lecture-notes/?utm_source=openai))
  5. Notes on the Combinatorial Fundamentals of Algebra by Darij Grinberg This detailed survey offers rigorous proofs and discussions on elementary combinatorics and algebra, including finite sums, binomial coefficients, and permutations. It's a treasure trove for those seeking a deeper theoretical understanding. ([arxiv.org](https://arxiv.org/abs/2008.09862?utm_source=openai))
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