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?

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

What distinguishes a permutation from a combination?

<p>Permutations consider order, while combinations do not. (C)</p> 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?

<p>12 (B)</p> 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)?

<p>3 (C)</p> 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?

<p>43758 (B)</p> Signup and view all the answers

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

<p>${ rac{n!}{r!(n-r)!}}$ (B)</p> 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?

<p>1800 (C)</p> 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?

<p>6C2 * 5C1 + 6C3 * 5C0 (B)</p> Signup and view all the answers

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

<p>Combinations do not consider the arrangement of objects. (C)</p> Signup and view all the answers

What does the notation nCr represent in combinatorial mathematics?

<p>total number of ways to select r objects from n objects (A)</p> Signup and view all the answers

Flashcards

Combination

The number of ways to choose a subset of r items from a set of n distinct items, without regard to order.

Permutation

The number of ways to arrange r items from a set of n distinct items, where order matters.

Combination Formula (nCr)

The formula for calculating the number of combinations of n objects taken r at a time. It is calculated as n! / (r! * (n-r)!).

Committee Problem

The way to determine the number of different committees that can be formed. In this case, order does not matter.

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Selection Problem

A problem where we need to determine the number of ways to select a group of items (often people) from a larger set, and the order of selection doesn't matter.

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nCr (Combination Formula)

The number of ways to choose r items out of n items, where order doesn't matter.

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nPr (Permutation Formula)

The number of ways to arrange r items out of n items, where order matters.

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Order is Not Important

In a combination, the order in which items are chosen doesn't affect the final result.

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Order is Important

In a permutation, the order in which items are arranged changes the result.

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Example 1: Choosing Leaders

Choosing a leader and an assistant leader from a group of 4 students is an example of a permutation, because the order in which they are chosen matters.

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Example 2: Choosing Representatives

Choosing two representatives from a group of 4 students is an example of a combination, because the order in which they are chosen doesn't matter.

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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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Related Documents

Combinations Math 10 PDF

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.

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