# Combinations are just an application of the counting principle?

• skrying
In summary, the conversation discusses the concept of combinations and permutations as applications of the counting principle. The counting principle is a method for determining the number of possible outcomes in an event involving multiple occurrences. The formula for combinations is N! / (K!(N-K)!) and for permutations is N! / (N-K)!. The conversation also thanks Robert for his explanation and examples, stating that they made the concept easier to understand.
skrying
Is it fair to say combinations are just an application of the counting principle? I already understand that permutations are just an application of fundamental principle and that combinations are just an application of permutations. If it's fair to say that combinations are in fact, just an application of the counting principle, then would their be a specific formula that proves as such? :!)

Pardon my ignorance. What is "the counting principle"?

The counting principle

The counting principle is dealing with the occurrence of more than one event, thus being able to quickly determine how many possible outcomes exist.
Kind of like sequences, if that makes more sense.

Counting principal for multiplication is: If something can be done in A ways and something else can be done in B ways, then the entire event can be done in AB ways.
For addition it means that in disjoint sets A and B, if we have K choices in A and L choices in B, then we have K+L choices in A union B.

Combinations and permutations then seem to be just that, applications of the counting principal. Possibly, skrying is aware that the combinations of N things taken K at a time is:

$$\frac{N!}{K!(N-K)!}$$ And for permutations: $$\frac{N!}{(N-K)!}$$

Last edited:
Thank you Robert

Thank you Robert, that explanation and the examples were really helpful. You explained it so it actually "makes sense" to me. Much appreciated!

## 1. How are combinations different from permutations?

Combinations involve selecting a group of objects without regard to order, while permutations involve arranging objects in a specific order. In other words, combinations focus on the selection of objects, while permutations focus on the arrangement of objects.

## 2. What is the counting principle used for in combinations?

The counting principle is used to determine the total number of possible combinations when selecting a certain number of objects from a larger group. It helps to ensure that all possible combinations are accounted for without any repetitions.

## 3. Can the counting principle be applied to any type of combination problem?

Yes, the counting principle can be applied to any type of combination problem, as long as the objects being selected are distinct and the order of selection does not matter.

## 4. How is the counting principle related to probability?

The counting principle is often used in determining the total number of possible outcomes in a probability experiment. By using the counting principle, we can calculate the probability of a specific combination occurring by dividing the number of desired outcomes by the total number of possible outcomes.

## 5. Can the counting principle be used for larger combination problems?

Yes, the counting principle can be used for larger combination problems involving a larger number of objects. However, it may become more complex and time-consuming, so it is important to break the problem down into smaller, manageable steps.

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