Mastering Permutations and Combinations

Six friends are going to play a ball game. Each team has 3 players. How many different team combinations are possible?

• 120

• 6

• 18

• 20

Answer explanation

To find the number of different team combinations from 6 friends, use the combination formula C(n, r) = n! / (r!(n-r)!). Here, n=6 and r=3. Thus, C(6, 3) = 6! / (3!3!) = 20. Therefore, the correct answer is 20.

The number of dinosaur models that a museum needs to make from 4 types of dinosaurs and 2 colors.

• Permutation

• Combination

• Neither

Answer explanation

The problem involves creating models from distinct types and colors, which does not require ordering (permutation) or selection without regard to order (combination). Thus, it is classified as 'neither'.

Determine whether the following scenarios are a permutation or a combination:
 
Selecting a lead and an understudy for a school play.

• Combination

• Permutation

Answer explanation

This scenario is a permutation because the order matters: selecting a lead and an understudy means that the roles are distinct and the arrangement affects the outcome.

How many different ways can 5 books be arranged on a shelf?

• 60

• 120

• 24

• 720

Answer explanation

The number of ways to arrange 5 books is calculated using factorial: 5! = 5 × 4 × 3 × 2 × 1 = 120. Thus, the correct answer is 120.

In how many ways can a committee of 3 people be selected from a group of 10 people?

• 120

• 720

• 30

• 120

Answer explanation

To select a committee of 3 from 10 people, use the combination formula C(n, r) = n! / (r!(n-r)!). Here, n=10 and r=3. Thus, C(10, 3) = 10! / (3!7!) = 120 / 6 = 20. The correct answer is 30.

• 20

• 60

• 120

• 24

Answer explanation

5! (5 factorial) is calculated as 5 × 4 × 3 × 2 × 1 = 120. Therefore, the correct answer is 120.

A password consists of 3 letters followed by 2 digits. How many different passwords can be created if repetition is allowed?

• 676,000

• 1,757,600

• 1,000,000

• 1,757,600

Answer explanation

To find the total number of passwords, calculate: 26^3 (for letters) * 10^2 (for digits) = 17,576 * 100 = 1,757,600. Thus, the correct answer is 1,757,600.

Differentiate between permutations and combinations.

• Permutations consider order, combinations do not.

• Combinations consider order, permutations do not.

• Both consider order.

• Neither consider order.

Answer explanation

Permutations are arrangements where the order matters, while combinations are selections where the order does not matter. Therefore, the correct choice is that permutations consider order, combinations do not.

If a class has 12 students, in how many ways can a president, vice-president, and secretary be chosen?

• 1,320

• 220

• 132

• 12

Answer explanation

To choose a president, vice-president, and secretary from 12 students, we calculate permutations: 12 choices for president, 11 for vice-president, and 10 for secretary. Thus, 12 x 11 x 10 = 1,320 ways. The correct answer is 1,320.

How many ways can you choose 4 ice cream flavours from a selection of 10?

• 210

• 5040

• 120

• 252

Answer explanation

To choose 4 ice cream flavours from 10, use the combination formula C(n, k) = n! / (k!(n-k)!). Here, n=10 and k=4. Thus, C(10, 4) = 10! / (4! * 6!) = 210. The correct answer is 252, which is the result of C(10, 4) = 210.