I fell asleep trying to figure this one out ... if you could help I would appreciate it ... Here is the problem ..

If Jon, Mac, and Heather are taking a group photo, how many different ways can the photographer line them up? .. okay, I have that one .. = 6 ... but now i have to add 3 more people and try to figure out how many ways can all of these together (6 of them)be lined up .... is there a formula i a could I use instead of writing all the different combinations? .. thank you for your help!

Wouldn't it be six factorial?

Yes, i understand that but can't figure out with what or how many different ways ..

Think of it as filling three spots with the three different people.

You can place one of 3 different people in the first spot.
Now look at the second spot, since one of the people has been placed, that would leave one of the remaining two people to be placed in that spot

So far you have 3*2 ways.

Finally since there is only one person left to placed in the last spot,
your calculations is 3*2*1 or 6 ways.

continued....

so if you had 6 people, repeat the thinking process and you have, as bobpursely told you,
6*5*4*3*2*1 = 6! = 720

To solve this problem, we can use the concept of permutations, which is the arrangement of objects in a specific order.

First, let's consider the first scenario where only Jon, Mac, and Heather are taking a group photo. In this case, there are three people, so the number of ways they can be lined up is simply 3!.

Now, let's consider the second scenario where three more people are added to the group. Since there are now 6 people in total, we need to find the number of ways they can be lined up. We can use the same logic as before and calculate 6!. However, because we are using the same three people from the previous scenario, we need to subtract the number of permutations we already counted.

Therefore, we have:

Number of ways to arrange all 6 people = 6! - 3!

To calculate this, we can use the factorial function:

6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
3! = 3 x 2 x 1 = 6

So, the number of different ways to line up all 6 people is 720 - 6 = 714.

In general, if you have n objects to arrange in a specific order, the formula to calculate the number of permutations is n!. However, if there are duplicates or some objects have already been arranged, you may need to adjust the formula accordingly.