Exam 6: Counting

There are 805 lockers in the athletic center and 4026 students who need lockers. Therefore, some students must share lockers. What is the largest number of students who must necessarily share a locker?

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Question 1

Correct Answer:

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By the generalized pigeonhole principle there are at least [4026/805] = 6 students who must share a locker.

How many students must be in a class to guarantee that at least ﬁve were born on the same day of the week?

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Question 2

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Since there are seven days of the week, to guarantee that at least ﬁve students were born on the same day requires at least 7 · 4 + 1 = 29 students.

How many rows are found in a truth table involving nine diﬀerent propositions?

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Question 3

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There are 29 = 512 rows in a truth table involving nine diﬀerent propositions.

How many diﬀerent license plates can be made if each license plate consists of three letters followed by three digits or four letters followed by two digits?

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Question 4

Find the value of each of the following quantities. (a) $C ( 5,4 )$ (b) $C ( 5,0 )$ (c) $P ( 5,1 )$ (d) $P ( 5,5 )$

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Question 5

What is the coeﬃcient of $x ^ { 2 } y ^ { 7 } \text { in } ( x + y ) ^ { 9 } ?$

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Question 6

(a) How many ways are there to choose 12 cookies if there are ﬁve varieties of cookies?

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Question 7

Show that $C ( n , r ) = C ( n , n - r )$ using (a) a combinatorial argument. (b) algebraic manipulation.

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Question 8

How many diﬀerent string can be made using all the letters in the word GOOGOL?

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Question 9

How many ways are there to choose ﬁve doughnuts if there are eight varieties (and only the type of each doughnut matters)?

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Question 10

(a) How many functions are there from a set with three elements to a set with eight elements? (b) How many one-to-one functions are there from a set with three elements to a set with eight elements? (c) How many onto functions are there from a set with three elements to a set with eight elements?

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Question 11

$\text { What is the coefficient of } x ^ { 7 } y ^ { 12 } \text { in } ( x + y ) ^ { 19 } \text { and in } ( 2 x + 3 y ) ^ { 19 } ?$

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Question 12

Each locker in an airport is labeled with an uppercase letter followed by three digits. How many diﬀerent labels for lockers are there?

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Question 13

(a) How many ways are there to arrange the letters of the word NONSENSE ? (b) How many of these ways start or end with the letter O ?

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Question 14

Exam 6: Counting

There are 805 lockers in the athletic center and 4026 students who need lockers. Therefore, some students must share lockers. What is the largest number of students who must necessarily share a locker?

How many students must be in a class to guarantee that at least ﬁve were born on the same day of the week?

How many rows are found in a truth table involving nine diﬀerent propositions?

How many diﬀerent license plates can be made if each license plate consists of three letters followed by three digits or four letters followed by two digits?

Find the value of each of the following quantities. (a) C(5,4)C ( 5,4 )C(5,4) (b) C(5,0)C ( 5,0 )C(5,0) (c) P(5,1)P ( 5,1 )P(5,1) (d) P(5,5)P ( 5,5 )P(5,5)

What is the coeﬃcient of x2y7 in (x+y)9?x ^ { 2 } y ^ { 7 } \text { in } ( x + y ) ^ { 9 } ?x2y7 in (x+y)9?

(a) How many ways are there to choose 12 cookies if there are ﬁve varieties of cookies?

Show that C(n,r)=C(n,n−r)C ( n , r ) = C ( n , n - r )C(n,r)=C(n,n−r) using (a) a combinatorial argument. (b) algebraic manipulation.

How many diﬀerent string can be made using all the letters in the word GOOGOL?

How many ways are there to choose ﬁve doughnuts if there are eight varieties (and only the type of each doughnut matters)?

(a) How many functions are there from a set with three elements to a set with eight elements? (b) How many one-to-one functions are there from a set with three elements to a set with eight elements? (c) How many onto functions are there from a set with three elements to a set with eight elements?

What is the coefficient of x7y12 in (x+y)19 and in (2x+3y)19?\text { What is the coefficient of } x ^ { 7 } y ^ { 12 } \text { in } ( x + y ) ^ { 19 } \text { and in } ( 2 x + 3 y ) ^ { 19 } ? What is the coefficient of x7y12 in (x+y)19 and in (2x+3y)19?

Each locker in an airport is labeled with an uppercase letter followed by three digits. How many diﬀerent labels for lockers are there?

(a) How many ways are there to arrange the letters of the word NONSENSE ? (b) How many of these ways start or end with the letter O ?

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

A: Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: Advanced Counting Techniques

Relations

A: Relations

Graphs

A: Graphs

Trees

A: Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

Find the value of each of the following quantities. (a) $C ( 5,4 )$ (b) $C ( 5,0 )$ (c) $P ( 5,1 )$ (d) $P ( 5,5 )$

What is the coeﬃcient of $x ^ { 2 } y ^ { 7 } \text { in } ( x + y ) ^ { 9 } ?$

Show that $C ( n , r ) = C ( n , n - r )$ using (a) a combinatorial argument. (b) algebraic manipulation.

$\text { What is the coefficient of } x ^ { 7 } y ^ { 12 } \text { in } ( x + y ) ^ { 19 } \text { and in } ( 2 x + 3 y ) ^ { 19 } ?$