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Combinatorics: A branch of mathematics that deals with counting methods, especially in complex situations.
It helps you organize choices and outcomes to avoid overcounting or undercounting.

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Example 1: Town Route Problem

3 roads from Ayre to Brantown

2 roads from Brantown to Cashcart

Total routes: 3 × 2 = 6

Includes a tree diagram showing all 6 options.

Example 2: Coin Flips

Flip a coin 3 times

Outcomes: H or T each time

Total possible results: 2 × 2 × 2 = 8

Tree diagram shows all 8 paths (HHH, HHT, ..., TTT)

Conclusion: Tree diagrams help visually organize all possible outcomes clearly.

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Example 1: Town Route Problem

3 roads from Ayre to Brantown

2 roads from Brantown to Cashcart

Total routes: 3 × 2 = 6

Includes a tree diagram showing all 6 options.

Example 2: Coin Flips

Flip a coin 3 times

Outcomes: H or T each time

Total possible results: 2 × 2 × 2 = 8

Tree diagram shows all 8 paths (HHH, HHT, ..., TTT)

Conclusion: Tree diagrams help visually organize all possible outcomes clearly.

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A problem solving technique for finding a quantity by determining another (opposite)quantity from which the first can be derived, in some situations, the indirect method makes calculations easier.

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Rule of Product:
If a task has multiple stages, multiply the number of options for each stage:
Total = m × n × p × …

Keyword: “AND”

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If one action can be performed in m ways and another action can be performed in n ways and the first and second action are mutualy exclusive,then there are m+ n ways in which either the first action or the second action can be performed.

The keyword for these problems is “ or "

To solve these problems, there are four steps:actions

1.Determne the number of actions

2.Determine if the actions are mutuauy exclusive

3.Determine the possblties for each action.

4. Add the number of possibilities together.

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Example 1:

You roll a 6-sided die twice. How many total outcomes?

Answer:
6 options for each roll → 6 × 6 = 36 outcomes

Example 2:

You choose a shirt (3 options) and a pair of shoes (2 options).
How many outfit combinations?

Answer:
3 × 2 = 6 combinations