Permutations, Combinations, and Fundamental Counting Principle

22 questions

All types22 Multiple choice10 Multiple select6 Numeric6

A technology committee has a pool of $8$ available guest speakers consisting of $5$ computer scientists and $3$ data privacy experts. A $4$ -person panel is to be formed from this pool. Which of the following statements regarding the possible panel selections must be true? Select all such statements.

Select all that apply

The total number of different

4

-person panels that can be formed without any restriction on specialty is equal to

70

The number of different

4

-person panels that consist of exactly

2

computer scientists and

2

data privacy experts is equal to

30

The number of different

4

-person panels containing at least

1

data privacy expert is equal to

65

The number of ways to select and arrange

4

computer scientists in a specific speaking order on stage is equal to

20

The number of different

4

-person panels containing no computer scientists is equal to

1

Show answer & explanation

Answer: The total number of different $4$ -person panels that can be formed without any restriction on specialty is equal to $70$ .; The number of different $4$ -person panels that consist of exactly $2$ computer scientists and $2$ data privacy experts is equal to $30$ .; The number of different $4$ -person panels containing at least $1$ data privacy expert is equal to $65$ .

Answer

The correct statements are those asserting that total unrestricted panels equal 70, panels with exactly 2 computer scientists and 2 privacy experts equal 30, and panels with at least 1 privacy expert equal 65.

The total unrestricted 4-person panels from 8 speakers is 8C4 = 70. Selecting 2 computer scientists (5C2 = 10) and 2 privacy experts (3C2 = 3) yields 10 × 3 = 30 panels. Using complementary counting, panels with at least 1 privacy expert equal total panels (70) minus panels composed entirely of computer scientists (5C4 = 5), giving 70 - 5 = 65.

Step-by-Step Solution

Calculate unrestricted combinations of 4 speakers out of 8

\binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70

Selection order does not matter for committee membership, so we use combinations.

Calculate combinations with restricted counts (2 CS and 2 PE)

\binom{5}{2} \times \binom{3}{2} = 10 \times 3 = 30

By the Fundamental Counting Principle, independent choices for each subgroup are multiplied.

Calculate combinations with 'at least 1' condition using the complementary counting method

Total panels minus panels with 0 PE:

70 - \binom{5}{4} = 70 - 5 = 65

Subtracting outcomes that violate the constraint from total possible outcomes simplifies 'at least 1' calculations.

Evaluate the permutation and impossible constraint statements

Arranging 4 CS out of 5 gives

P(5,4) = 120 \neq 20

. A 4-person panel with 0 CS requires 4 PE out of 3, which gives

0 \neq 1

Arrangement requires permutations, and panel requirements exceeding available pool members yield zero valid groups.

Key Concept

Combinations, Permutations, and Complementary Counting Principle

Question 22Question →

A librarian is arranging $7$ distinct books on a display shelf: $4$ history books and $3$ science books. If all $3$ science books must be placed adjacent to one another, in how many different linear orders can the $7$ books be arranged?

720

144

5,040

120

Show answer & explanation

Answer: 720

Answer

720 linear arrangements

To arrange items with an adjacency restriction, bundle the restricted items (the 3 science books) into 1 block. This leaves 4 history books and 1 science block, totaling 5 units to arrange linearly in 5! = 120 ways. Within the science block, the 3 distinct books can be arranged in 3! = 6 ways. By the Fundamental Counting Principle, the total number of linear arrangements is 5! × 3! = 120 × 6 = 720.

Step-by-Step Solution

Group the constrained items into a single block.

Treat the

3

distinct science books as a single unit or block, denoted as

[S]

Since the science books must always stand next to each other, they move as a single combined item.

Count the total number of items to arrange linearly.

We have

4

individual history books plus

1

science block, giving

4 + 1 = 5

items.

The

5

units can be ordered among themselves in

5!

ways.

Calculate the outer arrangements of the 5 units.

5! = 5 \times 4 \times 3 \times 2 \times 1 = 120

This accounts for all position permutations of the history books and the block.

Calculate internal permutations within the science block and multiply.

3! = 6

ways within the block. Total arrangements =

120 \times 6 = 720

By the Fundamental Counting Principle, total ways equals outer permutations multiplied by inner permutations.

Key Concept

Permutations with Adjacency Restrictions (Block Method)

PreviousPage 2 / 2

Permutations, Combinations, and Fundamental Counting Principle

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes