Chapter 10: Introduction to Counting

Content locked

The catalog is public. Reading this content requires approved access.

10.1 Sum and product rules

Introduction to counting

The product rule

Animation content:

Animation captions:

PARTICIPATION ACTIVITY

Check Show answer

Counting strings

The sum rule

PARTICIPATION ACTIVITY

Animation content:

Animation captions:

PARTICIPATION ACTIVITY

Start

Additional exercises

EXERCISE

10.2 The bijection rule

Number of people

Animation content:

Animation captions:

PARTICIPATION ACTIVITY

Check Show answer

Animation content:

Animation captions:

EXERCISE

{Abe, Ben, Cam, Don, Eli, Fran, Gene, Hal, Ike, Jan}

(Fran, Gene, Hal, Jan, Abe, Don, Cam, Eli, Ike, Ben)

10.3 The generalized product rule

Definition 10.3.1: Generalized product rule.

Animation content:

Animation captions:

EXERCISE

10.4 Counting permutations

PARTICIPATION ACTIVITY

Animation content:

Animation captions:

8 Employees:

Check Show answer

PARTICIPATION ACTIVITY

Animation content:

Animation captions:

PARTICIPATION ACTIVITY

Check

Show answer

CHALLENGE ACTIVITY

Start

Additional exercises

EXERCISE

10.5 Counting subsets

PARTICIPATION ACTIVITY

Using the k-to-1 rule to count subsets

Animation content:

Animation captions:

Animation content:

Animation captions:

PARTICIPATION ACTIVITY

Check

Show answer

PARTICIPATION ACTIVITY

Check

Show answer

Check

Show answer

Check

Show answer

Counting binary strings with a fixed number of 1s

of 5-bit strings with exactly 2 1s

of 2-subsets of {1, 2, 3, 4, 5}.

Animation content:

Animation captions:

PARTICIPATION ACTIVITY

Check Show answer

CHALLENGE ACTIVITY

Start

Additional exercises

EXERCISE

10.6 Subset and permutation examples

PARTICIPATION ACTIVITY

Animation content:

Animation captions:

Check Show answer

Additional exercises

EXERCISE

10.7 Counting by complement

PARTICIPATION ACTIVITY

Animation content:

Animation captions:

PARTICIPATION ACTIVITY

Check

Show answer

Check Show answer

CHALLENGE ACTIVITY

Additional exercises

EXERCISE

10.8 Permutations with repetitions

PARTICIPATION ACTIVITY

Animation content:

Animation captions:

PARTICIPATION ACTIVITY

Check Show answer

Example 10.8.1: Counting assignments using permutations with repetition.

Additional exercises

EXERCISE

10.9 Counting multisets

PARTICIPATION ACTIVITY

Animation content:

Animation captions:

Animation content:

Animation captions:

PARTICIPATION ACTIVITY

Fact 10.9.1: Counting multisets.

PARTICIPATION ACTIVITY

Explanation of solution (1:21).

PARTICIPATION ACTIVITY

Check

Show answer

Check Show answer

Additional exercises

EXERCISE

10.10 Assignment problems: Balls in bins

Figure 10.10.1: Six different balls-into-bins problems.

Figure 10.10.3: Balls in bins: At most one ball per bin.

PARTICIPATION ACTIVITY

Additional exercises

EXERCISE

10.11 Inclusion-exclusion principle

Animation content:

Animation captions:

PARTICIPATION ACTIVITY

Check Show answer

The inclusion-exclusion principle with three sets

Animation content:

Animation captions:

Animation content:

Animation captions:

Check

Show answer

Check

Show answer

The general inclusion-exclusion principle

PARTICIPATION ACTIVITY

CHALLENGE ACTIVITY

Start

The inclusion-exclusion principle and the sum rule

Determining the cardinality of a union by complement

Animation content:

Animation captions:

PARTICIPATION ACTIVITY

Check Show answer

CHALLENGE ACTIVITY

Start

EXERCISE

10.12 Counting problem examples

PARTICIPATION ACTIVITY

Animation captions:

selections with no restrictions - # selections \neg (\leq 3 jelly) OR \neg (\leq 5 maple). By De Morgan’s law.

selections with no restrictions - # selections (≥ 4 jelly OR ≥ 6 maple). Not

selections with no restrictions - [# selections (≥ 4 jelly) + # selections

Example 10.12.1: Solution to the license plate counting problem.

PARTICIPATION ACTIVITY

Additional exercises

EXERCISE

\{LPW, QKJ, CDP, USU, BBD, PST, LSA, RHR\}

EXERCISE

Source