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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?
A process of grouping objects into unordered sets.
A method for counting subsets regardless of order.
A selection of objects where order does not matter.
An arrangement of objects where order matters.
A permutation is an arrangement in which the order of the objects is significant. This distinguishes it from combinations, where order does not matter.
What is a combination?
A selection of objects where order does not matter.
A method for ordering objects into sequences.
An arrangement of objects where order matters.
A process that counts the number of sequences.
A combination involves choosing items from a set in which the order of selection is irrelevant. This concept is fundamental in combinatorics for counting selections.
How many distinct arrangements (permutations) does a set of n distinct items have?
2❿
n!
n
n²
The number of distinct arrangements of n distinct items is given by n! (n factorial). This is a basic and essential result in permutation counting.
What is the generating function for the sequence (1, 1, 1, 1, ...)?
1 - x
1/(1 - x)
1/(1 + x)
1 + x
The generating function for the sequence 1, 1, 1, ... is the sum of the power series x❰ + x¹ + x² + ... which converges to 1/(1 - x) for |x| < 1. This function is a staple in solving combinatorial problems.
Which statement best describes a recurrence relation?
It defines each term of a sequence based on its preceding terms.
It exclusively counts the number of permutations of objects.
It is used to generate power series for combinatorial sequences.
It provides an explicit formula for the nth term of a sequence.
A recurrence relation defines each term of a sequence in terms of one or more of its prior terms. This method is essential for solving sequences where a direct formula may not be readily available.
Using the Principle of Inclusion-Exclusion, how many integers from 1 to 100 are divisible by 2 or 3?
68
64
67
66
There are 50 integers divisible by 2 and 33 divisible by 3 in the range. Subtracting the 16 numbers divisible by both (i.e., by 6) using Inclusion-Exclusion gives 50 + 33 - 16 = 67.
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 + 2x)/((1 - x)(1 - 3x))
1/((1 - 3x)(1 - x))
(1 + x)/((1 - x)(1 - 3x))
Multiplying the recurrence by x❿ and summing over n yields the equation G(x)(1 - 3x) = 1 + 2x/(1 - x). Simplifying leads to the generating function G(x) = (1 + x)/((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
72
70
76
Count the numbers divisible by 2 (50), 3 (33), and 5 (20). Subtract the counts for intersections: divisible by 6 (16), 10 (10), and 15 (6), then add back those divisible by 30 (3). This calculation yields 50 + 33 + 20 - 16 - 10 - 6 + 3 = 74.
Which of the following best describes Pólya's Enumeration Theorem in counting colorings?
It solves recurrence relations by using generating functions.
It provides a method for counting restricted permutations.
It sums the total number of subsets of a set.
It counts distinct colorings by considering symmetry and group actions.
Pólya's Enumeration Theorem counts the number of distinct colorings by taking symmetries into account. This theorem prevents overcounting by considering the orbits of colorings under group actions.
What is the number of distinct arrangements of the letters in the word 'BANANA'?
180
60
120
360
The word 'BANANA' has 6 letters with repetitions (A appears 3 times, N appears 2 times, and B appears once). The number of distinct arrangements is given by 6!/(3!*2!*1!) = 60.
In a balanced block design, which parameter represents the number of blocks in which each treatment appears?
k
r
b
λ (lambda)
In the context of balanced incomplete block designs, the parameter r denotes the number of blocks in which each treatment appears. Understanding these parameters is key in design theory.
What is the coefficient of x❴ in the expansion of (1 + x)❷?
35
56
28
21
The binomial theorem indicates that the coefficient of x❴ in (1 + x)❷ is given by the combination C(7, 4), which equals 35. This is a direct and classic application of binomial coefficients.
Solve the recurrence relation a(n) = 2a(n-1) + 3 with a(0) = 1 and provide its closed form.
4·2❽❿❻¹❾ - 3
2❽❿❺¹❾ - 3
4·2❿ - 3
4·2❿ + 3
The recurrence a(n) = 2a(n-1) + 3 is solved by finding the homogeneous solution and a particular solution. Upon applying the initial condition a(0) = 1, the closed-form solution is determined to be 4·2❿ - 3.
Which generating function represents the partition function p(n), where p(n) counts the number of partitions of n?
((1 - x)/(1 - 2x))
∝ (from k=1 to ∞) [1/(1 - xᵝ)]
1/(1 - x)
∑ (from k=1 to ∞) [xᵝ/(1 - xᵝ)]
The generating function for the partition function is given by the infinite product ∝ₖ₌₝∞ 1/(1 - xᵝ). This form compactly encodes the number of partitions of an integer n and is pivotal in partition theory.
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?
4
10
8
6
Since one person is fixed in the committee, you only have to choose 2 members from the remaining 4 people. This selection can be made in C(4, 2) = 6 different ways.
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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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