Permutation and Combination

Today, we're diving into permutations! Can anyone tell me what a permutation is?

Isn't it just an arrangement of items where the order matters?

Exactly! For example, if we have three people, A, B, and C, how many ways can we arrange two of them?

I think it would be AB, AC, BA, BC, CA, and CB. That’s six arrangements!

Great job! We can calculate this using the permutation formula P(n, r) = n! / (n - r)!. Here, n is the total items, and r is how many we pick.

So what's n!?

Good question! n! stands for 'n factorial,' which is the product of all positive integers up to n. Let's say n = 3, then 3! = 3 × 2 × 1 = 6.

So for our case, it would be 3! / (3 - 2)! = 6 / 1 = 6. It all makes sense!

Wonderful! To remember, think of the phrase: 'P is for Position' to recall that order matters in permutations.

So, what’s the formula for permutations?

P(n, r) = n! / (n - r)!.

Now let's shift gears to combinations. Who can explain what this means?

It’s when the order doesn’t matter! Like choosing 2 fruits from an assortment.

Exactly! Can you provide an example with three fruits, say, apple, banana, and cherry?

Sure! The pairs would just be AB, AC, or BC. The order wouldn’t count, right?

Well done! We denote this by the combination formula C(n, r) = n! / (r! * (n - r)!).

Is n! still the factorial of n?

Yes! And r! is the number of ways to arrange the r selected objects. This accounts for order not being important.

So it’s the total arrangements divided by arrangements of what's selected?

Precisely! A helpful mnemonic for combinations could be 'C is for Choose,' as you are just selecting.

Can anyone remind me of the combination formula?

C(n, r) = n! / (r! * (n - r)!).

Let's explore what happens when we allow repetitions in permutations. What would be an example?

If we have three flavors of ice cream, can we pick two scoops where a flavor can repeat?

Exactly! If we have flavors A, B, and C, how would we calculate the number of different combinations for two scoops?

Umm, wouldn’t that just be 3 options for the first scoop and 3 for the second?

Great observation! Therefore, it would be 3^2 = 9 possible arrangements. The formula would be n^r for this case.

So, it's like multiplying choices for each slot!

Exactly! To remember, think of 'Repetition equals multiplication.' In worksheets, we approach problems considering whether repetition is allowed or not.

So how do we express our findings?

For permutations with repetition, n^r!

Now let’s examine combinations where repetitions are allowed. Can someone give an application of this?

How about choosing donuts? We can choose any number of the same type!

Exactly! For example, if you can select three donuts from four types, how would you visualize this?

I guess we’d picture the choices laid out with dividers between types of donuts?

Very well put! This method leads us to the formula for combinations with repetition, C(n + r - 1, r).

Could you clarify the ‘-1’ part in the formula?

Absolutely! In this case, we represent each chosen object with one cross and separate object types with a line, linking the cross and the total choices together.

What’s the overall significance of these tools in combinatorics?

These tools allow us to approach countless practical problems. Just remember, 'Combinations choose, permutations order.' That's an important takeaway!