# Combinations: An Introduction

- Definition of Combinations
  - Unlike permutations, combinations are selections where order does not matter.
  - Notation:
    - nPr: permutations of n distinct objects taken r at a time.
    - nCr: combinations of n distinct objects taken r at a time.

- Importance of Understanding Combinations
  - Essential for solving problems in probability, statistics, and combinatorial problems.

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# Understanding nCr

- Formula for Combinations:
  - The number of combinations of n objects taken r at a time is given by:
    \[
    nCr = \frac{n!}{r! \cdot (n-r)!}
    \]

- Application Example: Choosing Letters
  - How many ways to choose 2 letters from the set {A, B, C, D}?
  - {A, B} is the same as {B, A}, leading to fewer unique combinations.

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# Example: Choosing Students

- Problem:
  - How many ways can 3 students be chosen from 7 labeled as {A, B, C, D, E, F, G}?

- Calculation:
  - Each combination corresponds to 3! permutations.
  - Formula: 
    \[
    7C3 = \frac{7!}{3! \cdot (7-3)!}
    \]
  - Result: 35 ways to choose the students.

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# Visualizing Combinations with a Tree Diagram

- Tree Diagram Representation
  - Helps visualize the different arrangements that can be made when choosing objects.

- Description:
  - A tree diagram starting from a central point, with branches representing choices available at each step.

{The image of a tree diagram illustrating the choice of letters A, B, C, and D, with branches leading to different combinations such as AB, AC, AD, etc.}

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# Special Properties of Combinations

- Key Properties:
  - nC0 = 1 (one way to select no objects)
  - nCn = 1 (one way to select all objects)
  - nC1 = n (n ways to select one object)
  - nCn−1 = n (same as not selecting one object)

- Example Applications:
  - Selecting vertices for triangles from points on a circle.
  - Handshake problems in a group.

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# Solving Combination Problems

- Examples to Solve:
  - How many different pizzas can be made with 3 toppings from 9 options?
    - Calculation using 9C3.

  - Counting pairs from a larger set:
    - How many subsets of {1, 2, ..., 20} have exactly 2 elements?
    - Calculation using 20C2.

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# Using Calculators for Combinations

- Steps for Various Calculators:
  - TI-Nspire: Menu > Probability > Combinations
  - Casio: Run-Matrix mode; Probability menu

- Encouragement to Practice:
  - Use the calculator to solve:
    - 20C10 (selecting students in a class).

{The image of a calculator screen showing the computation of combinations using a TI-Nspire CX model.}

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This structured presentation covers the topic of combinations at a Year 11 level, providing foundational knowledge, examples, and interactive components to engage students. The descriptions for images are included to assist with visual aids that can enhance understanding.