In how many different ways can six of ten people be seated

in a row of six chairs?

I don't have the answer, but from my rough understanding of permutations and combinations, I got:

10C6 * 6! = 210 * 6!

I figure first 6 people have to be chosen out of the 10, and that can be done in 210 ways. Then, they can be seated in 6! different ways, so I multiplied the two quantities.

Thanks for any help!

Well, isn't that just a chair-ific question! Let's see if we can put a smile on your face with the answer.

First, let's address the number of ways to choose the 6 lucky people out of 10. You're absolutely right, that would be 10 choose 6, or 10C6, which is 210.

Now we're left with these rowdy six individuals who need to find their seats. And boy, do they have options! Since there are 6 chairs, the first person can sit in any of those 6 chairs. Once they're seated, the second person can plop down in any of the remaining 5 chairs, and so on.

So it's time to get our math boots on. If we multiply these numbers together, we get 6 x 5 x 4 x 3 x 2 x 1, which is 720. That's right, there are 720 different ways those six friends can take their seats.

But we're not finished yet! We need to multiply the number of ways to choose the people (210) by the number of ways they can be seated (720). So the total number of different ways those six lucky ones can be seated in that row of chairs is... *drumroll* 210 x 720, which equals 151,200.

So there you have it, my friend! A grand total of 151,200 ways to mix and match those six people in those six chairs. Have fun with all the seating arrangements!

You're on the right track! To find the number of different ways to seat six out of ten people in a row of six chairs, you need to use combinations, permutations, and multiplication to calculate the total number of possibilities.

First, you correctly calculated the number of ways to choose 6 people out of the 10. This can be done using the combination formula:

10C6 = 10! / (6! * (10 - 6)!) = 210

Now, since the chosen 6 people need to be seated in a row of 6 chairs, we need to calculate the number of ways to permute the chosen 6 people. This can be done using the permutation formula:

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

Finally, to get the total number of different seating arrangements, we multiply the number of ways to choose the people (210) by the number of ways to permute them (720):

Total number of arrangements = 210 * 720 = 151,200

Therefore, there are 151,200 different ways to seat six of ten people in a row of six chairs.

You're on the right track! To find the number of different ways in which six out of ten people can be seated in a row of six chairs, you need to use the concept of combinations and factor in the permutations within each combination.

Here's how we can calculate it step by step:

Step 1: Selecting the six people
Since you want to select six people out of ten, you can use the combination formula, denoted as "nCr" or "(n choose r)".

In this case, you want to calculate 10C6, which can be calculated as:

10C6 = 10! / (6! * (10-6)!)

Step 2: Permuting the selected people
Now that you have selected six people, you need to find the number of ways they can be seated in the row of six chairs. Since the order matters, you need to use the permutation formula.

In this case, there are six chairs, and six people to place, which can be calculated as:

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

Step 3: Calculating the final result
To get the total number of different seating arrangements, you need to multiply the number of ways of selecting the six people (combination) by the number of ways they can be seated (permutation).

So, the final calculation would be:

10C6 * 6! = (10! / (6! * (10-6)!) * 6!

Substituting the values we calculated earlier:

10C6 * 6! = (10! / (6! * 4!)) * 720

Simplifying further:

10C6 * 6! = (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((6 * 5 * 4 * 3 * 2 * 1) * 4!)) * 720

This calculation will give you the total number of different ways in which six of ten people can be seated in a row of six chairs.