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Combinations in Mathematics

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Questions and Answers

How many ways can the GILAS basketball team select 12 players out of 16?

140

4368

560

1820 (correct)

What is the total number of ways to form a committee of 3 teachers and 3 students from 5 teachers and 6 students?

300 (correct)

900

150

500

If a team of 5 must include at least 3 boys from 5 girls and 8 boys, how many ways can this team be formed?

1920

1360 (correct)

1040

1280

What is the formula for finding the combination of n objects taken r at a time?

$C(n, r) = \frac{n!}{r!(n - r)!}$ Signup and view all the answers

What distinguishes a permutation from a combination?

Permutations consider order, while combinations do not. Signup and view all the answers

How many different arrangements are possible when selecting a leader and an assistant leader from a group of 4 students?

12 Signup and view all the answers

How many combinations can be made by selecting 2 balls from a set of 3 balls (red, yellow, green)?

3 Signup and view all the answers

From a class of 18 students, how many ways can 8 students be selected for an overseas learning experience?

43758 Signup and view all the answers

What is the formula for finding the number of combinations when selecting r objects from n objects?

${rac{n!}{r!(n-r)!}}$ Signup and view all the answers

How many ways can a team of 2 boys and 3 girls be formed from 6 boys and 10 girls?

1800 Signup and view all the answers

If a committee of 3 members is to include at least 2 women, how is the total number of combinations calculated using 6 women and 5 men?

6C2 * 5C1 + 6C3 * 5C0 Signup and view all the answers

When order is not important, what is the primary characteristic that differentiates combinations from permutations?

Combinations do not consider the arrangement of objects. Signup and view all the answers

What does the notation nCr represent in combinatorial mathematics?

total number of ways to select r objects from n objects Signup and view all the answers

Study Notes

Combinations

Combinations are a way of selecting items from a set where the order does not matter.
In contrast, permutations consider the order of the items.

Key Differences Between Permutation and Combination

Permutation: Order matters (different order is a different outcome).
Combination: Order does not matter (different order is the same outcome).

Formula for Combinations

The formula for calculating combinations is: nCr = n! / ( (n-r)! * r!)
- n represents the total number of items.
- r represents the number of items being chosen.
- "! " represents factorial (e.g., 5! = 5 * 4 * 3 * 2 * 1)

Examples

Example scenario 1: Choosing a team of 2 students from a class of 4 where order does not matter. Answer is 6 different combinations possible.
Example scenario 2: Choosing a team of 3 teachers and 3 students from a class of 5 teachers and 6 students (where order does not matter).
Example scenario 3: Choosing 3 members from 6 women and 5 men which must include at least 2 women. Answer is 95 possible combinations.
Example scenario 4: Choosing 12 players from a team of 16 players for a basketball game. Answer is 18,564 possible combinations.

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Description

Explore the concept of combinations and learn how they differ from permutations. This quiz will cover the formula for combinations, including examples to solidify your understanding. Test your knowledge on selecting items from a set where order does not matter.