Combinations

Combinations are a method in combinatorics used to calculate the number of ways to select a group of items from a larger set without considering the order of selection.

Formula

The formula for combinations is:

C(n, r) = \frac{n!}{r!(n-r)!}

Where:

$n$ is the total number of items.
$r$ is the number of items to select.
$!$ represents factorial, which is the product of all positive integers up to that number (e.g., $5! = 5 \times 4 \times 3 \times 2 \times 1$ ). Combinations differ from permutations because the order of selection does not matter in combinations, whereas it does in permutations.

Steps to Calculate Combinations

Determine $n$ and $r$ : Identify the total number of items ( $n$ ) and the number of items to select ( $r$ ).
Apply the Formula: Substitute the values of $n$ and $r$ into the formula:

C(n, r) = \frac{n!}{r!(n-r)!}

Calculate Factorials: Compute the factorials of $n$ , $r$ , and $n-r$ , and simplify the expression.
Find the Result: Perform the division to find the total number of combinations.

Worked Examples

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Example 1: Choosing Committee Members

Problem: A club has $10$ members.

How many ways can they choose a $3$ -member committee?

Solution:

Step 1: Identify $n=10$ and $r=3$

Step 2: Apply the formula:

C(10, 3) = \frac{10!}{3!(10-3)!} = \frac{10!}{3! \times 7!}

Step 3: Simplify using factorials:

C(10, 3) = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = \frac{720}{6} = 120

Answer: There are 120 ways to choose the committee.

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Example 2: Selecting Students for a Group

Problem: A teacher has $15$ students and needs to select $5$ for a project.

How many ways can this be done?

Solution:

Step 1: Identify $n=15$ and $r=5$

Step 2: Apply the formula:

C(15, 5) = \frac{15!}{5!(15-5)!} = \frac{15!}{5! \times 10!}

Step 3: Simplify using factorials:

C(15, 5) = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1} = \frac{360360}{120} = 3003

Answer: There are 3,003 ways to select the group.

Summary

Combination Formula:

C(n, r) = \frac{n!}{r!(n-r)!}

Order does not matter in combinations.
Factorials are key to computing combinations.
Applications include selecting groups, teams, or subsets where order is irrelevant.

Combinations Simplified Revision Notes for Leaving Cert Mathematics

Combinations

Combinations

Formula

Steps to Calculate Combinations

Worked Examples

Example 1: Choosing Committee Members

Example 2: Selecting Students for a Group

Summary

500K+ Students Use These Powerful Tools to Master Combinations For their Leaving Cert Exams.

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