Possible Combinations Important Notes - Class 4 Mathematics Olympiad | Fully Solved Notes For Students

What Are Combinations?

A combination is a way of choosing or grouping items where we are concerned only with which items are chosen, not the order in which they appear. Combinations are like making a recipe with different ingredients or choosing what clothes to wear from your wardrobe - we care about which items are together, not which came first.

• Combinations count different possible groups or selections from a set of items.
• When order matters (for example, arranging medals as first, second, third), we use permutations. When order does not matter (for example, selecting team members), we use combinations.

• With objects: Choosing or grouping physical objects such as toys, fruits, shirts, or pencils.
• With numbers: Finding how many different groups of numbers give a target total, or choosing numbers from a set regardless of order.
• With or without repetition: Choosing where you may use the same item more than once (with repetition) or only once (without repetition).
• Order vs no order: If the order of items matters, it is a permutation problem; if it does not, it is a combination problem.

• List them out: For small sets, write each combination to ensure none are missed.
• Use the product rule: If you make a choice in stages and each stage is independent, multiply the number of choices at each stage.
• Use a tree diagram: Draw branches for each choice to visualise all possible combinations clearly.
• Use the combination formula (for older or faster work): For choosing r items from n items where order does not matter, the number is \(^{n}C_{r} = \dfrac{n!}{r!(n-r)!}\). This is usually introduced later, but is useful to know.

• Organise choices into stages: Break the selection into simple steps and count choices in each step.
• Look for patterns: Often combinations grow regularly; recognising patterns helps count quickly.
• Avoid double counting: Ensure the same group is not counted more than once when order does not matter.
• Use simple examples first: Try small numbers and list results to build confidence before solving larger problems.

Example 1 - Outfits (Shirts and Pants)

Problem: If we have three colours of shirts (red, blue, and green) and two pairs of pants (black and khaki), how many different outfits can we make by mixing and matching them?

Sol.

There are three choices of shirt and two choices of pants.

\(3 \times 2 = 6\)

Therefore, there are 6 different outfits in total.

Explanation (listing):

• Red shirt with black pants
• Red shirt with khaki pants
• Blue shirt with black pants
• Blue shirt with khaki pants
• Green shirt with black pants
• Green shirt with khaki pants

Example 2 - Choosing Colours (Order Does Not Matter)

Problem: From three colours, how many ways can we choose two colours if the order does not matter?

Sol.

List the possible pairs:

• Red and Blue
• Red and Green
• Blue and Green

There are 3 different combinations.

Alternative (using the combination idea):

Choosing 2 from 3 gives \(^{3}C_{2} = 3\).

Example 3 - Small Tree Diagram

Problem: You have 2 types of fruit (apple, banana) and 3 types of juice (orange, mango, grape). How many fruit+juice pairs can you make?

Sol.

Make choices in two stages: choose a fruit, then choose a juice.

There are 2 choices for fruit and 3 choices for juice.

\(2 \times 3 = 6\)

List to check if needed:

• Apple + Orange
• Apple + Mango
• Apple + Grape
• Banana + Orange
• Banana + Mango
• Banana + Grape

• Do not count arrangements where only the order changes when order does not matter. For example, {red, blue} is the same as {blue, red}.
• When choices are independent (like shirt and pant), multiply the number of options for each choice.
• If items cannot be repeated, remember to reduce the available choices after each selection when using listing or permutation ideas.
• Use small examples and lists to check answers if you are unsure, then generalise to larger problems.

Combinations help us count how many groups or selections can be made when order does not matter. Use listing, tree diagrams, or the product rule for simple problems. For larger problems, the combination formula \(^{n}C_{r} = \dfrac{n!}{r!(n-r)!}\) gives the count of ways to choose r items from n when order does not matter. Practise with small examples to understand the idea clearly.