Counting Principles

6 questions

All types6 Multiple choice4 Numeric1 Matching1

A 5-character security passcode must be created using the digits 0 through 9. The passcode must begin with an odd digit, end with an even digit, and no digit may be repeated anywhere in the passcode. How many different passcodes can be created?

1,400

8,400

18,000

25,000

Show answer & explanation

Answer: 8,400

Answer

8,400

To find the total number of possible passcodes, we evaluate the restricted positions first. The first digit must be odd (5 options: 1, 3, 5, 7, 9). The last digit must be even (5 options: 0, 2, 4, 6, 8). Because odd and even sets do not overlap, exactly one odd and one even digit are removed from the total pool of 10 digits. Because repetition is not allowed,

10 - 2 = 8

digits remain available for the middle three positions. The number of ways to fill the middle three spots is a permutation without repetition:

8 \times 7 \times 6 = 336

. Multiplying the independent choices for all positions gives

5 \times 336 \times 5 = 8,400

Step-by-Step Solution

Determine the number of options for the first character.

5 options

The first digit must be odd, and there are 5 odd digits (1, 3, 5, 7, 9) in the standard set of digits from 0-9.

Determine the number of options for the last character.

5 options

The last digit must be even, and there are 5 even digits (0, 2, 4, 6, 8) in the standard set.

Determine the number of available digits for the remaining three middle positions.

8 digits

There are 10 digits in total. Since no digit may be repeated and 2 distinct digits have already been used for the first and last positions, exactly 8 digits remain available.

Calculate the number of ways to arrange the middle three characters.

336 ways

Using permutations for the 3 middle spots from the 8 remaining digits: 8 × 7 × 6 = 336.

Calculate the total number of possible passcodes.

8,400 passcodes

Using the Fundamental Counting Principle, multiply the number of options for each stage: 5 × 336 × 5 = 8,400.

Key Concept

Fundamental Counting Principle with Restrictions

Question 2Question →

A military supply depot has $6$ different types of field rations, $4$ different types of hydration packs, and $3$ different types of medical kits. A soldier must build a standardized supply bundle containing exactly $2$ different types of field rations, $1$ hydration pack, and $1$ medical kit. How many different supply bundles can be created?

180

360

Show answer & explanation

Answer: 180

Answer

180

To find the total number of distinct supply bundles, we must calculate the number of ways to make each independent selection and multiply them together. First, calculating the number of ways to choose

2

rations from

6

requires combinations because order does not matter:

C(6, 2) = 15

. There are

4

ways to choose a hydration pack and

3

ways to choose a medical kit. Multiplying these together using the Fundamental Counting Principle yields

15 \times 4 \times 3 = 180

possible bundles.

Step-by-Step Solution

Determine the number of ways to select the field rations.

C(6, 2) = \frac{6 \times 5}{2 \times 1} = 15

ways.

Since the order in which the two rations are selected does not matter, we use the combinations formula.

Determine the number of ways to select the hydration pack and the medical kit.

4

ways to select a hydration pack and

3

ways to select a medical kit.

Selecting

1

item from a group of

n

items gives

n

possible choices.

Calculate the total number of possible supply bundles.

15 \times 4 \times 3 = 180

different supply bundles.

Apply the Fundamental Counting Principle by multiplying the number of independent choices together.

Key Concept

Combinations and the Fundamental Counting Principle

Question 3Question →

A communications center needs to assemble a rapid response team from a pool of $7$ qualified technicians. The team must consist of exactly $1$ team chief and $3$ field technicians. The $3$ field technicians share identical duties. How many different rapid response teams can be formed?

Show answer & explanation

Answer: 140

Answer

140

To find the total number of possible teams, we must account for the distinct roles across two stages of selection. First, we select

1

team chief from the

7

available technicians, which gives

7

options. Next, we select

3

field technicians from the remaining

6

technicians. Because the field technicians have identical roles, the order in which they are selected does not matter. This requires calculating a combination:

C(6,3) = \frac{6!}{3!(6-3)!} = 20

. Finally, using the Fundamental Counting Principle, we multiply the number of choices for the chief by the number of combinations for the field technicians:

7 \times 20 = 140

Step-by-Step Solution

Select the team chief.

7 possible choices

There are 7 available technicians, and 1 must be chosen for the distinct role of team chief.

Select the field technicians.

20 possible combinations

After choosing the chief, 6 technicians remain. We need to select 3 for identical roles. Since order does not matter, this is a combination: C(6,3) = (6 × 5 × 4) / (3 × 2 × 1) = 20.

Calculate the total number of unique teams.

140 teams

Using the Fundamental Counting Principle, multiply the independent choices for each stage: 7 × 20 = 140.

Key Concept

Combining the Fundamental Counting Principle with combinations for multi-stage selection.

Question 4Question →

A logistics officer needs to form a specialized transport convoy consisting of exactly $2$ cargo trucks and $1$ security vehicle. If there are $6$ available cargo trucks and $3$ available security vehicles, how many different combinations of vehicles can be selected for the convoy?

Show answer & explanation

Answer: 45

Answer

The total number of convoy combinations is found by calculating the combinations for each vehicle type and multiplying them together. Choosing

2

cargo trucks from

6

yields

15

combinations, and choosing

1

security vehicle from

3

yields

3

combinations. Multiplying these results gives

45

possible convoys.

Step-by-Step Solution

Calculate the number of ways to select the cargo trucks.

_{6}C_{2} = \frac{6!}{2!(6-2)!} = 15

Since the selection order does not matter, use combinations to choose

2

cargo trucks from the

6

available.

Calculate the number of ways to select the security vehicle.

_{3}C_{1} = 3

Use combinations to choose

1

security vehicle from the

3

available.

Determine the total combinations for the convoy.

15 \times 3 = 45

Apply the Fundamental Counting Principle by multiplying the independent category selections together.

Key Concept

Combinations and the Fundamental Counting Principle

Estimated Time:1m 30s

Question 5Question →

Match each counting scenario to the mathematical concept that is used to solve it.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The cleaning crew matches to Combination, the spelling bee winners matches to Permutation, and the lunch menu matches to the Fundamental Counting Principle.

Each scenario perfectly aligns with the core definition of its matching counting principle based on whether order matters, and whether the events are sequential and independent.

Step-by-Step Solution

Analyze the volunteer scenario to determine if order matters.

It matches to a Combination.

Because the members of a cleaning crew have identical roles, selecting Person A then Person B is exactly the same as selecting Person B then Person A.

Analyze the spelling bee scenario to determine if order matters.

It matches to a Permutation.

Since 1st, 2nd, and 3rd places are distinct, the arrangement or sequence of the winners matters.

Analyze the lunch menu scenario to identify the counting structure.

It matches to the Fundamental Counting Principle.

The scenario involves a sequence of independent events (choosing from different categories), which requires multiplying the number of choices.

Key Concept

Distinguishing between combinations (order does not matter), permutations (order matters), and the Fundamental Counting Principle (multiplying independent events).

Question 6Question →

A military dining facility offers $5$ different main courses, $3$ different side dishes, and $4$ different beverages. If a standard meal consists of exactly one main course, one side dish, and one beverage, how many different meal combinations can be chosen?

Show answer & explanation

Answer: 60

Answer

There are

60

different meal combinations possible.

The correct answer is

60

. According to the Fundamental Counting Principle, when there are multiple independent choices to be made to form a single outcome, the total number of combinations is found by multiplying the number of options for each choice. Here, we multiply the

5

main courses by the

3

side dishes and the

4

beverages:

5 \times 3 \times 4 = 60

possible meal combinations.

Step-by-Step Solution

Identify the number of available choices for each part of the meal.

Main courses =

5

, Side dishes =

3

, Beverages =

4

The Fundamental Counting Principle requires knowing the total number of options for each independent selection stage.

Apply the Fundamental Counting Principle by multiplying the choices together.

5 \times 3 \times 4

When selections are made in sequence and are independent of one another, the total number of combinations is the product of the number of options at each individual step.

Calculate the final product.

60

Multiplying the choices yields the total possible unique meal combinations.

Key Concept

Fundamental Counting Principle

Counting Principles

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Step-by-Step Solution

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Common Mistakes

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Step-by-Step Solution

Key Concept

Common Mistakes

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Step-by-Step Solution

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Common Mistakes

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Step-by-Step Solution

Key Concept

Common Mistakes

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Key Concept

Common Mistakes