With regard to Combinations, order matters. True False
The correct answer and explanation is:
Correct Answer: False
Explanation (Approximately 300 words):
In combinatorics, the distinction between combinations and permutations hinges on whether order matters.
- In combinations, order does not matter.
- In permutations, order does matter.
Let’s break it down:
A combination is a selection of items from a larger pool, without considering the order in which they are selected. For example, choosing 3 students out of a group of 10 to form a team is a combination, because the team members are not ranked or arranged—only the group itself matters.
The formula for combinations is: nCr=n!r!(n−r)!^nC_r = \frac{n!}{r!(n – r)!}
Where:
- nn is the total number of items,
- rr is the number of items to choose,
- !! denotes factorial, the product of all positive integers up to that number.
Let’s consider an example:
Suppose you are choosing 2 fruits from a basket containing Apple, Banana, and Cherry.
- Possible combinations (order doesn’t matter): AB, AC, BC. That’s 3 combinations.
- If you were counting permutations (order matters), you’d list: AB, BA, AC, CA, BC, CB — 6 total permutations.
Clearly, combinations count fewer possibilities because they treat, for instance, AB and BA as the same group, while permutations treat them as different.
This difference is critical in fields like statistics, probability, and computer science, where understanding how items are grouped can affect outcomes.
In summary, combinations ignore order, which is why the statement “With regard to Combinations, order matters” is False. Understanding this principle helps avoid common errors when solving problems involving selection or arrangement of items.