Free Practice Quiz Question List

07 Advanced Counting Methods Online Quiz Questions

Use this free practice quiz with 20 questions to review 07 Advanced Counting Methods, test your knowledge, and prepare for your next test or exam.

20 questions

Choose one

1 point

A committee of 5 people is chosen from a group of 10 people. How many different committees are possible?

A
50
B
100
C
200
D
252

True or false

1 point

For integers $0\le k\le n$ , the identity $\binom{n}{k}=\binom{n}{n-k}$ is true.

A
True
B
False

Choose one

1 point

How many integers from 1 through 100 are divisible by 2 or by 3?

A
50
B
66
C
67
D
83

Written response

1 point

How many nonnegative integer solutions does $x_1+x_2+x_3+x_4=6$ have?

Choose all

1 point

Select all statements that correctly follow from the binomial theorem.

A
The coefficient of $x^{n-k}y^k$ is $\binom{n}{k}$ .
B
The coefficient of every term is 1.
C
Setting $x=y=1$ gives $\sum_{k=0}^{n}\binom{n}{k}=2^n$ .
D
The theorem applies only when $n$ is negative.

True or false

1 point

A complete recurrence-based counting description requires initial conditions, a recurrence relation, and a justification that the cases are exhaustive and disjoint.

A
True
B
False

Fill in the blank

1 point

For binary strings with no consecutive 1s, complete the recurrence and the first initial condition: $a_n=$ , with $a_0=$ and $a_1=2$ .

Choose one

1 point

A $2\times n$ board is tiled with $2\times1$ dominoes, and $t_n$ denotes the number of tilings. What is $t_5$ ?

A
5
B
6
C
8
D
13

Written response

1 point

How many derangements are there of 4 objects?

Choose all

1 point

Select all correct statements about the signs in the general inclusion-exclusion formula.

A
Intersections of one set are added.
B
Intersections of two sets are added.
C
Intersections of three sets are added.
D
The sign for an intersection of $r$ sets is $(-1)^{r+1}$ .

Fill in the blank

1 point

For the recurrence $a_n=3a_{n-1}+2$ with $a_0=1$ , fill in the constant particular solution $p=$ and the closed form $a_n=$ .

Open ended

1 point

Solve the recurrence $a_n=2a_{n-1}+3$ with initial condition $a_0=1$ . Show how the homogeneous and particular solutions combine to produce a closed form.

Choose one

1 point

What is the coefficient of $x^3y^2$ in $(x+2y)^5$ ?

A
20
B
40
C
80
D
160

True or false

1 point

When deriving a counting recurrence by cases, the cases must be both exhaustive and disjoint.

A
True
B
False

Choose one

1 point

What is the coefficient of $x^3y^5$ in $(x+y)^8$ ?

A
28
B
56
C
70
D
112

Choose one

1 point

How many integers from 1 through 50 are divisible by 3 or 4?

A
20
B
22
C
24
D
28

Choose one

1 point

Let $a_n$ be the number of binary strings of length $n$ with no consecutive 1s, with $a_0=1$ and $a_1=2$ . What is $a_5$ ?

A
8
B
13
C
16
D
21

Choose one

1 point

How many selections of 6 objects can be made from 4 types when repetitions are allowed and order does not matter?

A
56
B
70
C
80
D
84

Written response

1 point

Given $b_n=b_{n-1}+n$ with $b_0=4$ , what is the value of $b_4$ ?

Written response

1 point

A software test suite contains 9 distinct modules. How many unordered sets of exactly 4 modules can be selected for a test group? Enter the exact number of selections.