Combinations (Leaving Cert Mathematics): Revision Notes

Combinations

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

1. Determine $n$ and $r$: Identify the total number of items ($n$) and the number of items to select ($r$).

2. Apply the Formula: Substitute the values of $n$ and $r$ into the formula:

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

3. Calculate Factorials: Compute the factorials of $n$, $r$, and $n-r$, and simplify the expression.

4. Find the Result: Perform the division to find the total number of combinations.

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Worked Examples

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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.