Question 1

Find $$\left| A _ { 1 } \cup A _ { 2 } \cup A _ { 3 } \cup A _ { 4 } \right| \text { if each set } A _ { i }$$ has 100 elements, each intersection of two sets has 60 elements, each intersection of three sets has 20 elements, and there are 10 elements in all four sets.

Accepted Answer

The answer of Find \[\left| A _ { 1 }...

Question 2

Use generating functions to solve $$a _ { n } = 5 a _ { n - 1 } + 3 , a _ { 0 } = 2$$

Accepted Answer

The answer of Use generating functions to solve \[a _...

Question 3

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes (labeled A, B, C) if envelope A has at least three coins in it.

Accepted Answer

The answer of Set up a generating function and use...

Question 4

Set up a generating function and use it to ﬁnd the number of ways in which nine identical blocks can be given to four children if each child gets at least one block.

Accepted Answer

The answer of Set up a generating function and use...

Question 5

Use generating functions to solve $$a _ { n } = 7 a _ { n - 1 } - 10 a _ { n - 2 } , a _ { 0 } = 1 , a _ { 1 } = 1$$

Accepted Answer

The answer of Use generating functions to solve \[a _...

Question 6

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for 0, 0, 0, a₀, a₁, a₂, . . . .

Accepted Answer

The answer of If G(x) is the generating function for...

Question 7

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for 0, 0, 0, a₃, a₄, a₅, . . . .

Accepted Answer

The answer of If G(x) is the generating function for...

Question 8

Use generating functions to solve $$a _ { n } = 3 a _ { n - 1 } + 2 ^ { n } + 5 , a _ { 0 } = 1$$

Accepted Answer

The answer of Use generating functions to solve \[a _...

Question 9

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a₀, 3a₁, 9a₂, 27a₃, 81a₄, . . . .

Accepted Answer

The answer of If G(x) is the generating function for...

Question 10

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a0, 0, 0, a₁, 0, 0, a₂, 0, 0, a₃, . . . .

Accepted Answer

The answer of If G(x) is the generating function for...

Question 11

Find the number of positive integers $\leq 1000 \text { that are multiples of at least one of } 3,5,11 \text {. }$

Accepted Answer

The answer of Find the number of positive integers \(\leq...

Question 12

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for 5, a₁, 0, a₃, a₄, a₅, . . . .

Accepted Answer

The answer of If G(x) is the generating function for...

Question 13

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a0, 0, a₁, 0, a₂, 0, a₃, 0, a₄, . . . .

Accepted Answer

The answer of If G(x) is the generating function for...

Question 14

Set up a generating function and use it to ﬁnd the number of ways in which nine identical blocks can be given to four children, if each child gets at most ﬁve blocks.

Accepted Answer

The answer of Set up a generating function and use...

Question 15

Suppose you use the principle of inclusion-exclusion to ﬁnd the size of the union of four sets. How many terms must be added or subtracted?

Accepted Answer

The answer of Suppose you use the principle of inclusion-exclusion...

Question 16

Find $$\left| A _ { 1 } \cup A _ { 2 } \cup A _ { 3 } \cup A _ { 4 } \right| \text { if each set } A _ { i }$$ has 150 elements, each intersection of two sets has 80 elements, each intersection of three sets has 20 elements, and there are no elements in all four sets.

Accepted Answer

The answer of Find \[\left| A _ { 1 }...

Question 17

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes (labeled A, B, C) envelopes A and B have the same number of coins in them.

Accepted Answer

The answer of Set up a generating function and use...

Question 18

Set up a generating function and use it to ﬁnd the number of ways in which nine identical blocks can be given to four children, if the oldest child gets three blocks.

Accepted Answer

The answer of Set up a generating function and use...

Question 19

Set up a generating function and use it to ﬁnd the number of ways in which nine identical blocks can be given to four children, if each child gets at least two blocks.

Accepted Answer

The answer of Set up a generating function and use...

Question 20

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a₃, a₄, a₅, a₆, . . . .

Accepted Answer

The answer of If G(x) is the generating function for...

Quiz 8: A: Advanced Counting Techniques

Find $\left| A _ { 1 } \cup A _ { 2 } \cup A _ { 3 } \cup A _ { 4 } \right| \text { if each set } A _ { i }$ has 100 elements, each intersection of two sets has 60 elements, each intersection of three sets has 20 elements, and there are 10 elements in all four sets.

Use generating functions to solve $a _ { n } = 5 a _ { n - 1 } + 3 , a _ { 0 } = 2$

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes (labeled A, B, C) if envelope A has at least three coins in it.

Set up a generating function and use it to ﬁnd the number of ways in which nine identical blocks can be given to four children if each child gets at least one block.

Use generating functions to solve $a _ { n } = 7 a _ { n - 1 } - 10 a _ { n - 2 } , a _ { 0 } = 1 , a _ { 1 } = 1$

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for 0, 0, 0, a₀, a₁, a₂, . . . .

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for 0, 0, 0, a₃, a₄, a₅, . . . .

Use generating functions to solve $a _ { n } = 3 a _ { n - 1 } + 2 ^ { n } + 5 , a _ { 0 } = 1$

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a₀, 3a₁, 9a₂, 27a₃, 81a₄, . . . .

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a0, 0, 0, a₁, 0, 0, a₂, 0, 0, a₃, . . . .

Find the number of positive integers $\leq 1000 \text { that are multiples of at least one of } 3,5,11 \text {. }$

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for 5, a₁, 0, a₃, a₄, a₅, . . . .

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a0, 0, a₁, 0, a₂, 0, a₃, 0, a₄, . . . .

Set up a generating function and use it to ﬁnd the number of ways in which nine identical blocks can be given to four children, if each child gets at most ﬁve blocks.

Suppose you use the principle of inclusion-exclusion to ﬁnd the size of the union of four sets. How many terms must be added or subtracted?

Find $\left| A _ { 1 } \cup A _ { 2 } \cup A _ { 3 } \cup A _ { 4 } \right| \text { if each set } A _ { i }$ has 150 elements, each intersection of two sets has 80 elements, each intersection of three sets has 20 elements, and there are no elements in all four sets.

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes (labeled A, B, C) envelopes A and B have the same number of coins in them.

Set up a generating function and use it to ﬁnd the number of ways in which nine identical blocks can be given to four children, if the oldest child gets three blocks.

Set up a generating function and use it to ﬁnd the number of ways in which nine identical blocks can be given to four children, if each child gets at least two blocks.

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a₃, a₄, a₅, a₆, . . . .

Quiz 8: A: Advanced Counting Techniques

Use generating functions to solve an=5an−1+3,a0=2a _ { n } = 5 a _ { n - 1 } + 3 , a _ { 0 } = 2an​=5an−1​+3,a0​=2

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes (labeled A, B, C) if envelope A has at least three coins in it.

Set up a generating function and use it to ﬁnd the number of ways in which nine identical blocks can be given to four children if each child gets at least one block.

Use generating functions to solve an=7an−1−10an−2,a0=1,a1=1a _ { n } = 7 a _ { n - 1 } - 10 a _ { n - 2 } , a _ { 0 } = 1 , a _ { 1 } = 1an​=7an−1​−10an−2​,a0​=1,a1​=1

If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for 0, 0, 0, a0, a1, a2, . . . .

If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for 0, 0, 0, a3, a4, a5, . . . .

Use generating functions to solve an=3an−1+2n+5,a0=1a _ { n } = 3 a _ { n - 1 } + 2 ^ { n } + 5 , a _ { 0 } = 1an​=3an−1​+2n+5,a0​=1

If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for a0, 3a1, 9a2, 27a3, 81a4, . . . .

If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for a0, 0, 0, a1, 0, 0, a2, 0, 0, a3, . . . .

Find the number of positive integers ≤1000 that are multiples of at least one of 3,5,11. \leq 1000 \text { that are multiples of at least one of } 3,5,11 \text {. }≤1000 that are multiples of at least one of 3,5,11.

If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for 5, a1, 0, a3, a4, a5, . . . .

If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for a0, 0, a1, 0, a2, 0, a3, 0, a4, . . . .

Set up a generating function and use it to ﬁnd the number of ways in which nine identical blocks can be given to four children, if each child gets at most ﬁve blocks.

Suppose you use the principle of inclusion-exclusion to ﬁnd the size of the union of four sets. How many terms must be added or subtracted?

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes (labeled A, B, C) envelopes A and B have the same number of coins in them.

Set up a generating function and use it to ﬁnd the number of ways in which nine identical blocks can be given to four children, if the oldest child gets three blocks.

Set up a generating function and use it to ﬁnd the number of ways in which nine identical blocks can be given to four children, if each child gets at least two blocks.

If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for a3, a4, a5, a6, . . . .

Use generating functions to solve $a _ { n } = 5 a _ { n - 1 } + 3 , a _ { 0 } = 2$

Use generating functions to solve $a _ { n } = 7 a _ { n - 1 } - 10 a _ { n - 2 } , a _ { 0 } = 1 , a _ { 1 } = 1$

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for 0, 0, 0, a₀, a₁, a₂, . . . .

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for 0, 0, 0, a₃, a₄, a₅, . . . .

Use generating functions to solve $a _ { n } = 3 a _ { n - 1 } + 2 ^ { n } + 5 , a _ { 0 } = 1$

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a₀, 3a₁, 9a₂, 27a₃, 81a₄, . . . .

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a0, 0, 0, a₁, 0, 0, a₂, 0, 0, a₃, . . . .

Find the number of positive integers $\leq 1000 \text { that are multiples of at least one of } 3,5,11 \text {. }$

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for 5, a₁, 0, a₃, a₄, a₅, . . . .

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a0, 0, a₁, 0, a₂, 0, a₃, 0, a₄, . . . .

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a₃, a₄, a₅, a₆, . . . .